Chapter 3 Bubbling in nonlinear elliptic problems near criticality

V p+I log(()~j)(N-2)/2Vj + (~,j)(N-2)/2(V -- Vj))

-~N-21~

RNUP+I -q-~

-q-fRN up+ll~ U + ~

Then we conclude

fs2

vp+l log V

8

- ~ N - 2 log(k1" 9")~k)(f~N ~p+l)+kfRN up+ll~ U + ~ hence from (2.25) and the previous computation we get

Ie(V) - Io(V) -- e (P +k I) 2 f~ u ~p+l _+_N - 2 l~ k p+l

fR ~ p + l l o g U ] + o ( e ) . u

"kk)(fRN~p-k-1) Vq

REMARK 2.1. The quantity o(e) in the expansion of (2.16) is actually also of that size in the C i-norm as a function of ~ and k in the considered region.

Bubbling in nonlinear elliptic problems near criticality

233

2.2. Variational reduction and conclusion of the proof In this section we prove Theorem 1.1. As mentioned in the previous section, in order to do so we first have to show the invertibility of the linear operator L (see (2.10)) subject to the orthogonality conditions given in (2.13). We will do it next for a slight modification of the linear operator L, which will be useful for the section devoted to the Brezis-Nirenberg problem. Indeed, we consider problem (2.13) with the linear operator L given by L(qS) -~ A 0

+/s

-+- (p + s)VP+C-149,

(2.26)

where # is less than # 1, the first eigenvalue of the Laplace operator with Dirichlet boundary condition on s We start with the resolubility of problem (2.13), keeping in mind that L is now given by (2.26). PROPOSITION 2.1. Let 6 > 0 be fixed. There are numbers so > O, C > O, such that, for points ~ in 22 and parameters )~ satisfying (2.17) problem (2.13) admits a unique solution 4) = T (h) for all 0 < s < so and all h ~ C ~ ($2e). Besides,

ttT(h)ll, <~Cllhll**

(2.27)

Icijl ~ Cllhll**.

(2.28)

and

PROOF. The proof of this proposition consists of 2 steps. STEP 1. Assume there exists a sequence e - Sn --+ 0 such that there are functions 4)E and

he with I[hsl[** = o(1), such that

0~ -=0

in f2E on a22E,

f Y2s gi p - 1 Z ij d/)s dx = 0

for all i, j,

I L(~E) = he + ~--~i,j cij Vip-1 Z i j

for certain constants Cij, depending on s. Then

We shall establish first the slightly weaker assertion that

114~,llp -

sup Ico-(r x EX2s

+

[a)-(I~-P+I/(N-2))DdDs(X)I ~

0

with p > 0 a small fixed number. To do this, we assume the opposite, so that with no loss of generality we may take ll4~slip - 1. Testing the above equation against Zlh, integrating

M. del Pinoand M. Musso

234 by parts twice we get that

E Cij fs'2s Vip-1 ZijZl -- fs2 [AZlh + (p + 8)wP-l+SZlh]r 8

-+"lZ82/(N-2) fs26 ~s Zlh - fsef hs Zlh 9 This defines a linear system in the since we have, for h = 1 . . . . . N,

s Vip-1 zij zlh = ~i,l~jh'

(2.29)

Cij which is "almost diagonal" as s approaches zero,

(

N v--PA?l OUAi,Ooxh + O(l )

(2.30)

and for h = N + 1,

Ai fn viP-1ZiJ ZI(N+I) -- ~i,l~J,N+I fR uP-I(xU-A' + (N s

2)UA~) 2 + o(1)

N

(2.31)

for suitable

Ai > O.

On the other hand, it is easy to see that, for 1 = 1 . . . . . k, we have [AZth + (p + s)Vp+s-IZlh] <~Co)(N+2)/(N-2)(X) for h -- 1 . . . . . N and [AZth + (p+ e)VP+S-IZlh] Co) (u+3)/(u-2) (x) for h -- N + 1. Hence, we get

fS2 [zXZlh+

(P + e)VP+~-IZlh]r

(2.32)

- o(1)114,~ lip,

8

after noticing that AZlh + PVlp-1Zlh -~ O, and an application of dominated convergence. Let us now estimate the second term in the right-hand side of (2.29). If N ~> 4 and h -- N 4- 1, then

/Z82/(N-2)

C82/(N-2) 114,~lip fs~ c~ 8

682/(N-2)

114'~lip f n ~ (1 -+- Ix - 1 ~~ l)N-p

68(2-P)/(N-2)

I1r lip.

Hence, in this case we conclude that, provided p is small,

IZE2/(N-2) fs ~sZlN+l = o(1)llr 8

If N ~> 4 and h = 1 . . . . . N, similar computation yields to

/Z82/(N-2) if2 ~)sZlh 8

-

-

O(1)E2/(N-2) ll~s llp - o(1)ll~b~ lip.

!

(1 4- Ix -- sej l) N-2

Bubbling in nonlinearellipticproblemsnearcriticality

235

Assume now that N = 3 and h = N + 1. Then

/Z62

C~21l~ellp f ~ ~ (1 -+-Ix -1 ~jl) 2-p ~< Cll4,~llpe

fx_2 ~eZlN+ 1

1-p

Hence we conclude that, provided p is small,

1Z62fl2 ~eZlN+l

-

-

o(1)l14,~llp.

8.

Analogously, one proves that, for N = 3 and h = 1 . . . . . N,

lZ62 is2 ~eZlh

-

o(1)llcP~llp.

E

Finally, for the last term in the right-hand side of (2.29), we have

/x?~.he Zlh

Cllh~ll**.

Thus, we conclude that

Ic~jl ~

(2.33)

Cllh~ll** + o(1)llqS~ll~

so that cij = 0(1). Taking into account that # < # 1, we can rewrite the equation in the following form

dpe(x) - (p+ e) fs ? Ge(x, y)VP+e-ldpedy- lze2/(N-2) fs ? Ge(x, y)dpe g

=-f~

Ge(x,y)hedy-Zcijfs?

ViP-lzijGe(x,y)dy,

c

x6X?e, (2.34)

where Gc denotes Green's function of I2E. Furthermore, the function qSe is of class C 1 and axj~be(x)

- (p + ~) fx? OxjGe(x, y)VP+e-lc/)edy

-- lZ62/(N-2)

Ge(x, y)dpe

-- - fs? Ox~Ge(x, y)he dy - Z cij fs? Vip-lZijOx~ G~(x, y) dy, x E s

(2.35)

M. del Pino and M. Musso

236

We make now the following observation

fs2 ]Gs(x, y)hsldY <~ IIh~ll**f fR F ( x - y)m4/(N-2)(x)dy s N Cllhsll**(Z(1-+-]x-~;[2)-(N-2)/2) i

ft.

Indeed, one has

s

k F(X-- y)oo4/(N-2)(x)dy <~ C ff-~s

-- y)(1-+- ly--~512) -2 dy

j=l k

C Z(Ilj j=l

-Jr-I2j -~-I3j),

where

Ilj -- fB(x,lx-~l/2) /2j -- L(seS,ix_,eSi/2)

F(x - y)(1 + [ y - ~j. 12)-2 dy, F(x - y)(1 + lY - ~.}le) -e dy

and

I3j -- f~i~N F ( x

y)(1 + lY - ~12) -2 d y - Ilj - I2j.

-

Now, (2.36) follows from

C llj <~ (1 -Jr-]XC-- ~j ]2)2 folX-~j l/2 t dt ~< (1 + Ix - ~j! I2) tN-1 122 <<.(1 + IX -- ~C 12)(N-2)/2 f01x-~} 1/2 (1 + t2) 2 dt

<<.

c

(1 + IX -- ~:j 12)(N-2)/2

C (1 + Ix - ~jl 2)

<<. and 13j ~ C[2j.

t(t 2 + 1) (N-6)/2 dt

(2.36)

m

~

c~

Jr

~ . ~

+

m

I

t~

~

+

9

~

~

+

I

+

M

m m

-~-

Jr

c~

~ .

I

~

~"

~

~ u

I

/A

I

~

~ ~

+

I

m.m

m..

/A

+

~

~~

~-

9

/A

~ . ~

I

Jr

/A

~

~

9

~~

~~

238

M. del Pino and M. Musso

Let us now estimate lZ62/(N-2) fx2e Go(x, y)¢e. First assume that N/> 4. Then

/ZE2/(N-2) ff2a, [ae(x'

y)4'~ [

<
i

~"

Ix - yl N-z (1 +

ly - ~/I) 2-p dy

= Ce2/(N-2)lncbellP~i (fB(x,lx_~fl/2)+ fB(~[,ix_~[i/2) + fs~\(B(x,lx-~[l/2)UB(*[,lx-~[I/2))) = CJ/¢N-Z)IlCo~IIp(A+ B + C). Now, first we have

l

~2/~N-2) IIq~ II,oA <~ Ce 2/~N-2) [Lee ttp ~

f Ix-g;[I/2 t dt

(1 + lY - ~[I) 2-p .Jo 1

1

~< c I1¢~II,o~-~ (1 + lY -~[I) 2-'°" i

Second,

1

8 2/(N-2) 114~llpB ~< c 82/(N-2) IlqSelip ~

flx-~[i/2 t N-3+p dt

(1 -F[y - ~ [ [ ) u - 2 J1

i 82/(N-2) Ix - ~[I N-2+p ~< Cl}¢~llp ~--~

i

(1 _t_ ix _ ~[I)N_ 2 1

~< Clle~ll~ ~

(1 + ly - ¢[I) z-p" i

Taking into account that the integral C can be estimated by the integral B, we conclude that, for N >/4,

tzs2/(N-2) fs?~]Ge(x, Y)q~e] ~< cII4~llpo)~-P(x).

Bubbling in nonlinearellipticproblems near criticality

239

If N -- 3, then one has

IGs(x' y)cks[ <~CIl~llpJ(x).

1Z82 fs 8

Similarly, one gets

J;

lZ82

{

cII4,~II,o09fi-P+I/ (N-2) (X)

[axjGs(x, Y)~sl <~ ClldPellpco~+l/(N-2)(x)

i f N >~4, if N = 3 .

8

Equations (2.34) and (2.35) and the above estimates imply that

{ c(114,~11,o+ IIh~ll**)J-'~ ]4'*(x)l ~< C(ll4,,llp + IIh, ll**)J(x)

i f N >~ 4,

(2.37)

i f N -- 3

and

{ C(ll4,~llp + Ilh~ll**)co~-p+l/(N-2)(x) c(ll ll + llh ll**)J+'(x)

i f N >/4,

(2.38)

if N = 3 .

In particular, we have that

O)-(fi-2P) (x)[~s(X)] ~ Co)P (x). Since p is arbitrarily small and limbslip - 1, it follows the existence of a radius R > 0 and a number y > 0, both independent of s such that [[4}s1l/~(gR(~[)) > Y for some i. Assume this happens for i -- 1. Then local elliptic estimates and the bound (2.37) yield that, up to a subsequence, q~s(x) = q~s(x - ~ [ ) converges uniformly over compacts of ]Ru to a nontrivial solution q~ of

+ pV--p-1 ,o

(2.39)

-o

for some A > 0, which besides satisfies

]~(X)[ ~ Clxl (2-N)(fi-p).

(2.40)

Hence, for N -- 3 we have [q~(x)[ ~

C[xl2-N.

Now, since q~ satisfies (2.39) and estimate (2.40) holds, a bootstrap argument leads to

Iqb(x)] ~ Clxl 2-N

for any N > 3.

M. del Pino and M. Musso

240

It is well known that this implies that ~b is a linear combination of the functions

OUA,o OXj

'

x 9 VUA,o -q- (N - 2)UA,0, see for instance [78]. On the other hand, we recall that

fl2

qbe Vip-1Zij = 0

for all i, j.

8

By dominated convergence, this relation is easily seen to be preserved up to the limit, hence

N

OXj

=

L

(bUA,o--P-1(x. VUA,0 + (N - 2)U--A,0) = 0

N

for all j. Hence the only possibility is that ~b =- 0, which is a contradiction which yields the proof of 114~elip ~ 0. Finally, from estimate (2.37), we observe that

hence II~ II, ~ O, and the proof is thus complete. STEP 2. Now we are in a position to prove Proposition 2.1. To do this, let us consider the space

Zij qb -- 0 Vi, j } endowed with the usual inner product [4~, ~ ] -

fs2~ V4~V~P. Denote with (f, g) the in-

ner product in LZ(~e), namely (f, g) = fs2~ fg for any f, g 6 L2(g2e). Problem (2.13) expressed in weak form is equivalent to that of finding a 4~ 6 H such that ItS, ~ ] = ((/Z82/(N-2)q~ -Jr-(p -Jr-8)gP+e-ldp - h), ~)

V~ G H.

With the aid of Riesz's representation theorem, this equation gets rewritten in H in the operational form 4~ - K(~b) + ft

(2.41)

with certain h 6 H which depends linearly in h and where K is a compact operator in H since /z
[

At~ @/Z82/(N-2)t~ @ (p @ 8) V P - I + ~ 4~=0 on 0~2~,

(~, Vip-1Zij ) -- 0

-- Z i , j cij Vip-1Zij

in S2E, (2.42)

241

Bubbling in nonlinear elliptic problems near criticality

for certain constants Cij. Assume it has a nontrivial solution 4~ = 4~e, which with no loss of generality may be taken so that ll4~e[I, = 1. But this makes the previous step applicable, so that necessarily 114~eI[, --+ 0. This is certainly a contradiction that proves that this equation only has the trivial solution in H. We conclude then that for each h, problem (2.13) admits a unique solution. We check that

II~ II, ~< c IIh I1,,. We assume again the opposite. In doing so, we find a sequence he with IIh~ I1.. = o(1) and solutions 4~e 6 H of problem (2.13) with 114~11, - 1. Again this makes the previous step applicable, and a contradiction has been found. This proves estimate (2.27). Estimate (2.28) follows from this and relation (2.33). This concludes the proof of the proposition. 5 It is important for later purposes to understand the differentiability of the operator T with respect to the variables ~' 6 S-2~, )~ 6 Rk+ which satisfy constraints (2.17). Consider the L ,~ (resp. L ~ ) of functions defined on I-2e with finite ]l" II, norm (resp. I[" ]l** norm). We consider the map (2.43)

(~',)~, h) w-> S(~', ,k, h) -- T(h), as a map with values in L ,~ N Hd (S-2e). We have the following result:

PROPOSITION 2.2. Under the conditions of Proposition 2.1, the map S is of class C 1 Besides, we have

II

s

z, h) ll, -< c IIh It**.

PROOF. Let us consider differentiation with respect to the variable ~hl, h - 1 . . . . . k, 1- 1 N For notational simplicity we write 0 -- 0~,. Let us set, ~b -- S(~' L, h) and, , ..., . O~ij ' still formally, Z = 0~,q~. We seek for an expression for Z. Then Z satisfies the following equation:

A Z -Jr-lzg2/(N-2) Z --[- (p + e) V P+e-I Z

= --(P + E)0~' (wP-l+e) ~b+ Z dij Wip - 1 Z i j -[--cijO~,(Wi p - 1 Z i j )

in s

i,j Here

dij = O~,cij.

Besides,

from

differentiating

(d/), gi p - 1 Z i j ) -- 0 we further obtain the relations

(~b, 0~.,(Vi P-1Zij))m E (Z,

Vi p-1Zij) - O.

Let us consider c o n s t a n t s bij such that

Z - ~ blhZlh, Vi p-1 Z i j ) -- O. l,h

the

orthogonality

condition

M. del Pino and M. Musso

242 These relations amount to

Z blh(Zlh, Vip - 1 Z i j ) - (d/), Of, Vip-1Zij ). l,h

(2.44)

Since this system is diagonal dominant with uniformly bounded coefficients, we see that it is uniquely solvable and that

blh - o(11r uniformly on ~t, ~ in the considered region. Now, we easily see that

Ilu,

I1..

cllu, ll..

Recall now that, from Proposition 2.1, r

I


=

O(llh iN**). On the other hand,

- ~/I -N-4,

hence

I[Cij Of, Vip- 1Zij l]** ~ C l]h II**. Let us now set rl -- Z - Y~-i,j bij Zij. Then, summing up the estimates above and using that [i4)I[, ~< C ]lh ll**, we get that r/satisfies the relation

/kT] -~-//82/(N-2) r] -Jr- (p -+- 8)V p-I+e rl -- f + ~

dij viP-I zij

in s

(2.45)

i,j where

f -- Z b i j ( I ( / k

-Jr-1282/(N-2) -Jr-(p + 8)VP-I+~))Zij -Jr-cijOf,(giP-lzij)

i,j - (p + e)Of,(Vp-I+~)$,

(2.46)

so that ]]f ]]** ~< C I]h ]]**. Since besides r/6 Hd (,f2e) and

(17, Vip-1Zij) -- 0

for all i, j,

we have that r / - T (f). Reciprocally, if we now define

Z -- T ( f ) + ~ , bij Zij i

i,j

(2.47)

Bubbling in nonlinearellipticproblems near criticality

243

with bij given by relations (2.44) and f by (2.46), then it is a matter of routine to check that indeed Z -- 0~,r In fact, Z depends continuously on the parameters ~ , A and h for the norm JJ. JJ,, and IJZJ[, ~< CJJhJJ** for points in the considered region. The corresponding result for differentiation with respect to the )~i's follow similarly. This concludes the proof. [-1 REMARK 2.2. We can also state the above result by saying that the map (~,)~) w-~ T is of class C ] in s176 , L , ) and, for instance,

(2.48)

(D~,T)(h) = T ( f ) + ~ b i j Z i j ,

i,j where f is given by (2.46) and

bij by (2.44).

Let us now go back to problem (2.2) and consider # = 0. Next step in the proof of Theorem 1.1 is the finite-dimensional reduction: we consider the nonlinear problem of finding a function r such that for some constants r the following equation holds

I A(V --[-r -]- (W -[- dp)p+e -- Ei,j cij wip-1 zij

in S2e, on 0~e, for all i, j.

r

fs2~ ckVip- 1Zij

- 0

(2.49)

Let us rewrite the first equation in (2.49) in the following form

Ar -q- (p -[- E) VP+e-lr -- - R e - Ne(r -[- ~

cij Wip-1 Zij

in ;2e,

i,j where, in this case, Re and Ne (q~) are defined respectively by (2.11) and (2.12). To estimate the JJ" JJ**-norm of Ne (r/), it is convenient, and sufficient for our purposes, to assume JJr/JJ, < 1. Note that

Ne (~7)-

(P +

E)(p 2

1 + E)

(V1 -+ V2-+-to)p-2+eO 2

(2.50)

with t 6 (0, 1). If N ~< 6 then p ~> 2, and we can estimate

]V-4/(N-2)Ne(I]) I ~ CV(P-2)fl-4/(N-2)+2~ ]]17]]2' hence

IIN ('7) I1., ctl ll ,. Assume now that N > 6. In the region where dist(y, 0Y2e) ~> •6 -1/(N-2) for some 6 > 0, then V (y) ~> ot~ V (y) for some c~ > 0; hence in this region, we have

[V-4/(N-2)Ne(r]) ] ~ CV2/3-1 ]]?]112~ C62/3-11J~Jl2

244

M. del Pino and M. Musso

On the other hand, when dist(y, OS2e) ~< 66 -1/(N-2), the following facts occur: V(y), V(y) = O(e) and, as y --+ O~2e, V(y) = Ce (N-1)/(N-2) dist(y, O~e) + o(e). This second assertion is a consequence of the fact that

2 V(y) = e 1/2 ~

UAjel/(N_2),~j (61/(N-2) y)

j=l and hence, taking into account that the Green function of the domain S2 vanishes linearly with respect to dist(x, 0 S2) as x ~ 0 S2,

V(y) - Ce dist(el/(N-2) y, OS2) + o(e) -- e (N-1)/(N-2) dist(y, OI2e) + o(e). These facts imply that, if dist(y, OI2e) ~< Se -1/(N-2) , then

IV-4/(N-2)Ne(o)I <~ V-4/(N-2)Vp-2IIrlII2

CV-4/(N-2) (6 (N-1)/(N-2) dist(y, 0$'2e)) p-2 dist(y, O~-2e)2 ll Orl(y) ll2 C6 -4/(N-z)+((N-1)/(N-z))(p-z)-p/(N-z)+z~+z/(N-2)

11/7112~ Ce 2~-1111/ll2

If 101 ~ 1 ~ then relation (2.50) yields that IV-4/(N-2)Ne(ll)I ~ CV2~-]llr/ll2 ~ Ce2~-1110112. In the other case, we see directly from (2.50) that IN~(~)I ~ Clrll p and hence IV-4/(N-2) ge(rl) l ~ VP~-4/(N-2) IIrI[IP ~ Ce-(2-P)/~IIITIIP. Combining these relations we get i f N ~<6,

CIIoll

i f N > 6.

(2.5])

Next we estimate the term Re. We have

IR l

Ivi

- v/ l +

~<

I log Vilr +

o(~(N+2~/(N-2~)

in the regions where Ix - ~[I ~< ~6-1/(N-2), for small ~ > 0. Taking into account that IRel ~< Ce (N+2~/(N-2) in the complement of these two regions, we get IIe~ll** ~ C~.

(2.52)

Bubbling in nonlinear elliptic problems near criticality

245

PROPOSITION 2.3. Assume the conditions of Proposition 2.1 are satisfied. Then there is a C > O, such that for all small e there exists a unique solution

r162

=~+~

to problem (2.49) with 7t defined by q/= - T ( R ~ ) .

(2.53)

Besides, the map (~', )0 --+ r I!r

k) is of class C 1for I1" II.-norm and

~< cc,

IIV(~,,x)r

(2.54) ~< ce.

(2.55)

PROOF. Problem (2.49) is equivalent to solving a fixed point problem. Indeed r - r + ~p is a solution of (2.49) if and only if

~)----- -T(N~((b + ~)) =--Ae((b) taking into account that ~ = - T (Re). Then we need to prove that the operator A~ defined above is a contraction inside a properly chosen region. First observe that, from the definition of ~p, from (2.52) and from Proposition 2.1, we infer that II1/,11., <~ c e and, by (2.51), for IIo II, ~< 1,

IIu~(* +')11.,

{ c(ll~ll~+ ~)

~<

i f N ~<6,

C(e2~-111~112 + e-(2-p)~ll~ll p + ep~+l)

i f N > 6.

(2.56)

Let us set

From Proposition 2.1 and (2.56) we conclude that, for e sufficiently small and any r/~ f'r, we have

[IA~(o)l[,

=

IIz-,~(N~(o + ~)) II. <~ cllN~(~ + o) I1..

{

C82~8 C(8 2fi+l + 8 p3+l) ~ 8

i f N ~<6, if N > 6,

M. del Pino and M. Musso

246

where the last inequality holds provided that e is sufficiently small. Now we will show that the map Ae is a contraction, for any e small enough. That will imply that Ae has a unique fixed point in ~ and hence problem (2.49) has a unique solution. For any 01, 02 in Ur we have

I1.,,

IlA~(ol)- A~(O2) ][, < CIIN ( + 01)m NS(~ -t-

hence we just need to check that Ne is a contraction in its corresponding norms. By definition of Ne, ,,p+e-1 DoN~(rl) = (p + e)[(V + -q)+

m V p+e-1

].

Hence we get ] N e ( ~ -1- 01) - N e ( ~ H" 02) I ~ CVp-21O[101 - 021

for some ~ in the segment joining 7t + 01 and 7t + 02. Hence, we get for small enough II0 II,, V-4/(N-2)[Ne(lP

@ 01) - N e ( ~ .qt_02)1 ~

C~-2~-111011,1101_

0211,.

We conclude

V 2fl- 1 (11rll II, -+- II/72II, + IIap II,) IIrll ~<

- 02

II,

e2~-1(110111, + 110211, + II?tll,)llrll - 0211, s 1101 - 0211,

i f N ~<6,

ifN >6,

and hence Ae is a contraction mapping for the II 9 II,-norm inside f r Let us now analyze the differentiability properties of the function r We recall that r is defined through the relation B(~',/.,,) =r Write N(~', )~, r -- Ne (r

)~).

T(N~(,+ ~ ) ) = 0 . namely

N(~', &, q~) - ( V + q~)P+e - V p+e - ( p + e)VP+e-l(b. Then

D~N(~', )~, (b) = (p + e)[(V + ~)~+~-1 _

gp+e-1]

and

D~, N (~', I., (b) = (p + ~)[(v + ~)~+~-1

-- g p+e-1 - ( p -~- 6 - 1) g P + e - 2 ~ ]

D~, V,

(2.57)

Bubbling in nonlinear elliptic problems near criticality

247

similarly for Dx N (~', X, dp). We have that

DcB(~',X, 4))[0] = 0 + T(OD~Ns(r + 7t)) = 0 + M(O). Now,

IlM(O) II, CIIODc N (r + I1,, ~ CIIv-4/(N-2)+~D4~N~(r + IIo11o11,. We have

V-4/(N-2)+fi]D~Ne(r + ~)1 ~< C

vmfi-1 I1r + ~P II, 2~-1 114' + ~P II,

if N <~ 6, if N > 6,

C8 min{2fi' 1} .

It follows that for small s, the linear operator DcB(~',X, dp) is invertible in L,~ , with uniformly bounded inverse. It also depends continuously on its parameters. Now, let us consider differentiability with respect to the (~', X) variables. We have

D~,B(~' X, ,4)) - (D~,L)(Ne(dP + O)) + [L((D~,N)(~', X, ~ + ~p)) + L((D~N)(~', X, q~+ ~p)D~,~)]. Here D~,L is the operator defined by the expression (2.48) and the second quantity by (2.57). Observe also that

D~,~ -- (D~, L)(Rs) 4- L(D~, Re).

(2.58)

D~[ Rs = (p + s) V p+e-1D~[ V1 - p~(-1D~[ V1.

(2.59)

Also

These expressions also depend continuously on their parameters. We have a similar expression for the derivative with respect to A. The implicit function theorem then applies to yield that r indeed defines a C 1 function into L ~ . Moreover, we have for instance

O~,dt) -- -(OcbB(~, )~, d/)))-I [(D~, L)(Ne(~ Jr- ~))

+ [L(D~,[N(~',X, dp+ ~)]) + L((D(~N)(~', X, ch + O)V~,~t)]].

M. del Pino and M. Musso

248

Hence,

+ IID,'N(~', x, r + V') I1.. + IID~sN(~', z, r + r

I1..),

where we have used Remark 2.2. From (2.56), we get

Ce

if N ~< 6,

(2.60)

CF_pfl+l if N > 6.

[[Ne(r + 7t)[]** ~<

On the other hand, from (2.57) we have

I(D~,N)(~', X, r [ c ~ ( N - 1 ) / ( N - Z ) I ( V _q_.~)p+e-l+ { V5/(N-Z)+e+/~llq~ll, C

65/(N_2)+e+flllr

_ gp+e-1

--

(p

-I- 8 --

1)VP+e-2r

i f N ~< 6, if N > 6.

Hence,

l[(D~'N)(~', X, ~ + r I1.. < ci1r RII, ~< ce. In similar way we get that

I]D~kN(~',~,, ~ + r

~
Hence, we finally get

IID~,r

~ C~,

as desired. A similar estimate holds for differentiation with respect to the ~,i'S. This concludes the proof. D For what we are going to do next, it is more convenient to recast the parameter ~,i into the parameters A i given by

~,N-2--aNA2

(2.61)

with

aN --

l

f R N u p§

(2.62)

p + 1 (fRN ~p)2"

With this in mind, let us consider points (~, A) which satisfy constraints ]~i -- ~j] > 6,

dist(~i, 0~2e) > 6,

6 < Ai < 6 -1,

Bubbling in nonlinear elliptic problems near criticality for some small fixed 6 > 0. Let r

A ( V %-r --[-(V + r r fs2~ r Vip- 1Zij - 0

= r p+e

249

A)(y) be the unique solution of problem

1 -- Z i , j cij Vip- Zij

in ~2e, on OY2~,

(2.63)

for all i, j

given by Proposition 2.3. Let us consider the functional j (~, A) = I~ (V + r

(2.64)

where Ie was defined in (2.15). The definition of r yields that i~'(v + r

= o

for all ~ which vanishes on OS2e and such that

fI2

17gi p-1Zi j _ 0

for all i, j.

It is easy to check that

OV OV O~ij ---- Zij + o(1), OAi

=

Zi(u+l)

@

o(1),

with o(1) small as e --+ 0. This fact, together with the last part of Proposition 2.3, give the validity of the following lemma. LEMMA 2.2. v = V + r is a solution of problem (2.2), namely cij

-

-

0 in (2.63)for all i, j,

if and only if (~, A) is a critical point of J. Next step is then to give an asymptotic estimate for J (~, A). We see next that this functional and Is(V) coincide up to order o(s). PROPOSITION 2.4. We have the expansion,

J(~, A) -- kCN -[- 6[yN + ll)NtItk(~, A) -Jr-o(1)],

(2.65)

where o(1) --+ 0 as e --+ 0 in the uniform Cl-sense with respect to (~, A) satisfying (2.17). Here,

qxk(~ , A ) - - - ~

EH(~j j--1

,

~ j ) A 2j - 2 E G ( ~ i
i~ ~j)AiAj

+log(A1...Ah)

,

(2.66)

250

M. del Pino and M. Musso

and the constants o)N and CN are those in Lemma 2.1 and

k

k

p + 1 (-ON + ~09N log a N

~3N --

k p+l

fR ffp+llogff} u

with aN given by (2.62).

PROOF. We start showing that J(~, A) - I s ( V ) -- o(s)

(2.67)

V$,A [J(~, A ) - I s ( V ) ] -- o ( e ) .

(2.68)

and

Taking into account that 0 = D I s ( V + 7t + ~)[~], a Taylor expansion gives 1

I s ( V + @) - I (~, A) --

foIt dt --

L

t dt D21s(V + ~ + tqb)[(b, (b],

IVq~l2 - (p + s ) ( g + 7t

+ t~)P+s-lqb 2

g

/o1

]

Ns (~ + O ) ~

t dt

(2.69) Since I1~11,-O(~), we get J(~, A ) -

Is(V + ~)-

(2.70)

0(62).

Differentiating with respect to ~ variables we get form (2.69) that D~[Ie(V+#,)-J(~,A)] -- e -1/(N-2)

L 1t at (L

DU [(Ne (q~ + 0))q~] E

+ (p + ~) f~ D~,[((V + ~, + t~)r+~-I E

-(v

+ o)~+~-'1~]). (2.71)

Bubblingin nonlinearellipticproblemsnearcriticality

251

Using the computations in the proof of Proposition 2.3 we get that the first integral in relation (2.71) can be estimated by O(E2), so does the second; hence De [J(~,

A) - Z~:(V+ ~)] = 0(62-1/(N-2)).

Now, since DZe` (V)[lk] = fs~ Re.0,

ze`(v + ¢~) -

ze`(v)

Since I t ' l l , + I[Re`[l** = O(e), the above term is O(e2); then (2.67) follows from (2.70) and (2.72). Using again (2.72), we see that

D~[Ze`(V + ~) -Ze`(V)]

=e-1/(N-2)O~,

11o'

(1 -- t)dt

E L ((V+tO)P+e`-I-VP+e`-I)o 2] (p+e)

c

Since from Proposition 2.3 it follows that IIO f , ~ II, = O ( e ) , we get

D~[I~.(V + O) -

2-e`(V)]

The desired result will follow if we prove that e -1/(N

2)D~,(f Re`~k)= o @ ) .

(2.73)

\ d £2E

First, if N > 3, Proposition 2.3 yields

g-1/(N-2)Du( f Re`~)= \d f2 e

0(~ 2-1/(N-2))

if N = 4, 5,

O(~7/4{ 1ogel)

if N = 6,

O(E(N+I)/(N-2))

if N ~> 7.

Let us consider now the case N = 3. We have that

D~, ( f~ R~fO = f~ (De[Rs) f" + fs2 (De, f' ) Re`= e2(I + II)"

252

M. del Pino and M. Musso

Let us estimate first H. Our first observation is that, locally, around ~ , e -1R~(~I + x) --+ V~ log Vo + cV 4 uniformly over compacts, for certain constant c. Here V0(Ixl) = Uz,0 for some )~ > 0. We also set Z0 - x . VV0 + V0. Hence, e-17t(x + ~l) ~ w(Ixl) where w is the unique radial solution of

A w 4- p V~) w = Vo5 log VO 4- c V~) 4- b V4 Zo which goes to zero at cx~, and is such that

finN V4 Z O1JJ-- O. The constant b is precisely that making the integral of the right-hand side of the above equation against Z0 equal to zero. In a similar way,

-1D$ l ~(x + ~:1) --->

x w'(Ixl)Ixl

After a suitable application of dominated convergence, we get that

// -- 6-2 fs2 (D~, ~)Re

log Vo +

+ bV4Zo)(ixl)m'(l

l)N - o,

by symmetry. The term I can we dealt with in a similar manner. We conclude that I --+ 0. Hence relation (2.73) has been established, and this proves the result in what concerns to derivatives with respect to ~. Derivatives with respect to A can be handle in a simpler way, since the term e -1/(N-2) does not appear in the differentiation. The validity of (2.73) thus follows. From Lemma 2.1, we can finally conclude that

J(~,

A ) -- k e N Jr- 8[yN + WNII/k(~,

A)] -~-o(~).

(2.74)

On the other hand, as a consequence of (2.68) and Remark 2.2 we also get VJ(~, A ) - - e ~R( fp + l

N

U p + I ) (Vt/'zk(~:' A ) + o(1)).

The proof is complete. We are now ready to give the proof of Theorem 1.1.

(2.75) D

Bubbling in nonlinear elliptic problems near criticality

253

PROOF OF THEOREM 1.1. For simplicity we give the proof of the theorem only for the supercritical case. As mentioned before, the subcritical case works exactly the same. Lemma 2.2 guarantees that v = V 4- ~b, where V is given by (2.7) and 4) is the solution given by Proposition 2.3, is a solution to problem (2.2) if and only if the point (~, A) is a critical point for J (~, A) -- I, (V + 4)) (see (2.64)). Hence we need to find a critical point for J, or equivalently, a critical point for J~(~, A ) -- CON[e -1 (J(~, A ) - k C N ) - fiN]

(see Proposition 2.4). Proposition 2.4 implies that (2.76)

J(~, A ) - q/k(~, A ) -- o(1),

where o(1) is in C 1 s e n s e as e --+ 0 and qJk is given by (2.66). The assumption of Theorem 1.1 is that qJk has a nontrivial critical point situation in 7). Hence, taking into account (2.76), there exists an ~ > 0 such that, for any e 6 (0, ~), there exists a critical point ( ~ , Ae) in 7) of J(~, A) such that VqJk(~, A~) ---> 0 as e --+ 0. The qualitative behavior of the solution predicted by Theorem 1.1 follows directly by construction from the definition of V and of ~b. E] REMARK 2.3. In the slightly subcritical problem (1.5), existence of bubbling solution is governed by existence of nontrivial critical point situations for ~k- (see (1.8)). Here we want to be more precise with the examples of contractible domains described in the Introduction on which problem (1.5) admits respectively multiple-bubbling solutions at one point and bubbling solutions at multiple points. These examples are contained in [65]. The first example is the following: take s - if21 U f22, where a'21 and a'22 a r e two smooth bounded domains such that ~21 A Y22 "-- ~. Assume that

~Q1 C {(Xl,

X') E IR X

RN-

I0 < a

Xl ~ b /

and

x')

•

I- b

< o/.

For any 6 > 0 let

c~ -/(xl,

x') ~ I~ • ~ u - a l xa ~ ( - b , b),

Ix'l ~ ~}

Let ~2~ be a smooth connected domain such that ~o C ,(2s c ~o U C~.

(2.77)

Let G o and H o denote respectively the Green function relative to a set D and its regular part. It is straightforward to show that lim Hs2~(x) = Hs20 (x)

8--+0

(2.78)

254

M. del Pino and M. Musso

C 1-uniformly on compact sets of S2o, and lim Gs2~ (x, y) - Gs20 (x, y) 6--+0

(2.79)

C 1-uniformly on compact sets of 120 x 120 \ {x = y}. Hence the number of nontrivial critical point situations of H ~ can be estimated from above by the sum of the numbers of nontrivial critical point situations of Hs~l and of Hs22. Observe now that both Hs21 has a strict and Hs~2 have a strict minimum point respectively in S-21 and in S'22. Furthermore, a strict minimum point is an example of nontrivial critical point situation. Let I-2~ be defined as in (2.77). By (2.78) we deduce that if 6 is small enough H ~ has two different strict minimum points and the claim is proved. A more general example with several disjoint sets linked together by a thin tube can be built with minor changes. We next construct an example of contractible domain where (1.5) admits a bubbling solution with multiple, say k > 1 bubbles. Let 120 -- 121 tO ... U I2k, where, $21 . . . . . S2k are k smooth bounded domains such that ~-i f-) -~-j -- ~ if i ~ j. Denote by q~ff0 the function q/k- relative to 1-20. It is easy to check that this function has a strict minimum point in the connected component ~1 • "'" • l-2k x Rk+ of the set ,f2ok x Rk+. Assume that ~f'2i C {(Xl, X t) E R • RN-1l ai <~Xl

~ bi

}

with

bi < a i + l ,

i

--

1. . . . .

k.

For any 6 > 0 let C~ = {(Xl, x') E R x

RN-I[xl ~ (al, bk), Ix'l ~< ~]-

Let I2~ be a smooth connected domain such that I20 C I2~ C $20 U Ca. As before we can prove that lim Hs2~(x) -- Hs2o (x) ~--~0 C 1-uniformly on compact sets of ~20, and lim Gs2~ (x, y) -- Gs20 (x, y) 8-+0 Cl-uniformly on compact sets of G0 x S20 \ {x = y}. Therefore we deduce that !Pk- relative to S2~ converges C 1-uniformly on compact sets of ~2~ x (R+) ~ to !P~ ~ Using again that a strict minimum point is an example of nontrivial critical point situation, we can conclude that if 6 is small enough the function q/ff~ has a strict minimum point, and hence problem (1.5) has a solution with k bubbles in 12~.

Bubbling in nonlinear elliptic problems near criticality

255

3. Solvability of slightly supercritical problems and the topology of the domain In this section we describe several concrete situations where problem (2.1) has a bubbling solution, or equivalently, where the function tPk (see (2.66)) does have a nontrivial critical point situation. It is here where the topology of the domain plays a fundamental role: it guarantees the existence of nontrivial critical point situation for qJk and hence a bubbling solution for (2.1).

3.1. The case of a small hole The first result we want to show is the case in which the domain s has a hole. Coron [24] proved that the problem at the critical exponent, namely when s - 0 in (2.1), has a solution whenever the domain 12 has a sufficiently small hole. We can prove that the same result holds true for the slightly supercritical case: namely when s > 0 in (2.1) is sufficiently small and 12 is a domain with a sufficiently small hole, problem (2.1) has a solution. Not only that: we can describe this solution. It is a two-peak solution, whose maximum points are located close to the little hole dropped. Let us mention two facts. First, it is hopeless finding a solution which exhibits just one peak, as rigorously proved by [11]. Roughly speaking, this is due to the fact that the function ~ --+ H(~, ~) cannot have a negative critical point, since it always assumes positive values. Second, the solution we find for the slightly supercritical case cannot be obtained from Coron's solution by a perturbation argument, since our solution uniformly converges to 0, as s --+ 0, on compact sets of I2 minus two points. The first result of this section is the following. THEOREM 3.1. Assume that

n-D\~o,

(3.1)

where 79 and co are bounded domains with smooth boundary in R N , N >~ 3, with the property that co C B (0, p) C 79. There exists a Po > 0 such that, ifO < p < Po is fixed and 12 is given by (3.1), then there exists ~ > 0 such that, f o r any 0 < s < ~, problem (2.1) has a solution us. Furthermore, there are two points ~1 and ~2 in 12, with ~1 =/: ~2, such that, f o r any 6 > 0 and as s --+ O,

sup

us(x) --+ O

u

x~B(~j,5)

and

sup

Us(X) "~ 8 - 1 / 2

V j -- 1, 2.

x~B(~j,6)

In view of Theorem 1.1, the proof of Theorem 3.1 consists in setting up a min-max scheme to find a critical point of the function qJ2 and to show that this is a nontrivial critical point situation.

256

M. del Pino and M. Musso

The function q/2 takes the explicit form t//2(sel, ~2, A1, A2) -- 1ATM(~)A + log(A1A2) 2

(3.2)

A T M ( ~ ) A -----H(sel, ~I)A12 + H(~2, se2)A 2 - 2 G ( ~ l , ~ 2 ) A 1 A 2

(3.3)

m

where

for ~ = (~1, ~2) E S2 • $2 \ {~1 = ~2}, A = (A1, A2) E R +. In order to show that ~2 has a critical point, we first need to prove some properties of the Green function and its regular part. Let 79 be a smooth bounded domain in I~ u containing the origin. We shall emphasize the dependence of Green's function on the domain by writing it as GT~(x, y), and similarly for its regular part HT~(x, y). Let us consider a number p > 0 and the domain m

79p = 79 \ B(O, p). We denote by Gp, Hp respectively its Green's function and regular part. LEMMA 3.1. The following result holds lim Hp(x, y) = HD(x, y),

p--+O

m

uniformly on x, y in compact subsets of 79 \ {0}. PROOF. The maximum principle yields

Hp(x, y) <<,HT~(x, y), hence the family of functions Ha(x, y) is uniformly bounded as p --+ 0 on each compact subset of 79 \ {0} x 79 \ {0}, and strictly increasing in p. By harmonicity, its pointwise limit as p --+ 0 is therefore uniform on compacts of 79 \ {0}. Since the resulting limit H(x, y) is harmonic in x and bounded, it extends smoothly as a harmonic function in all of 79. H therefore satisfies equation

AxH(x, y) --0,

x E 79,

H(x, y)-- F(x - y),

x~079,

and is thus equal to HT~.

E]

Consider now our domain 1-2 given by (3.1). Denote by G and H its Green's function and regular part, and consider the function go(se) -- qg(~l, se2) = H1/2(~el, sel)H1/2(~2, ~2) - G(~:I, ~2)

(3.4)

Bubbling in nonlinear elliptic problems near criticality

257

defined on I2 • I2 \ {~1 = ~2}. The sign of the function qg(~) is related to the sign of the determinant of the matrix M(~) defined in (3.3). Next two lemmas provide properties of the function q9 when S2 is given by (3.1), which will be useful in the sequel. More precisely, the first lemma says that, if the hole in S2 is sufficiently small, then there is a region around the hole where q9 assumes negative values; the second lemma says that, if the points (~1, ~2) are sufficiently far away from the boundary of I-2 then one can find a direction along which the gradient of q9(~l, ~2) is not zero. LEMMA 3.2. For any (fixed) sufficiently small n u m b e r cr > 0 there is a Po > 0 such that if o9 is any domain with co C B (0, p) and p < Po, then

sup q9(~1, ~2) < 0. 1~11=1~21=o PROOF. We have that HT~ is smooth near (0, 0) while GD becomes unbounded, hence for any cr > 0 sup

~(~1, ~2) < 0,

[~11--1~21 -'O"

where q3 is defined by

~ /4 1/2 qg(~l, ~2) -" H ; / 2(~1, ~1)--79 (~2, ~2) -- G/)(~I, ~2)On the other hand, for this or, it follows from the previous lemma that H and hence G become uniformly close to H~ and G ~ on [~11-- [~21-- O as /9 gets smaller. The desired conclusion then readily follows. Vq Let now ;2~ -- {x 6 s

dist(x, O~2) > 3 }. We have the lemma.

LEMMA 3.3. Given c < 0 there exists a sufficiently small n u m b e r ~ > 0 with the following property: I f (~1, ~2) E 0(S23 • f28) is such that qg(~l, ~ 2 ) - c, then there is a vector r, tangent to O(I2~ • I2~) at the point (~1, ~2), so that _

_

# o.

(3.5)

The n u m b e r 6 does not depend on c.

PROOF. Consider, for small 8, the modified domain

~ -- ~-1~C2, and observe that for this domain, its associated Green's function and regular part are given by

G(Xl, x2) - s N - 2 G ( S x l , Sx2),

/~(Xl, X2) -- ~ N - 2 H ( ~ X l , ~X2).

258

M. del Pino and M. Musso

T h e n ~ ( S x 1 , 6 x 2 ) -- c translates into q~(Xl, x2) = ct~N-2, w h e r e

q~(Xl, x2) -- H 1 / 2 ( X l , X l ) H 1 / 2 ( x 2 , x2) - G ( x l , x2). Assume that dist(3xl, O ~ ) -- 3, namely that dist(xl, O ~ ) - 1. After a rotation and a translation, we assume that the closest point of the boundary to Xl is the origin, that x l - (0, 1), where 0 - 0~u-~ and that as 6 --+ 0 the domain ~2 becomes the half-space x u > 0. In order to make the relation q3(xl, x2) - c~ u - 2 remain, as 6 --+ 0, we claim that necessarily we must have d - Ix l - xzl = O(1) as 3 --+ 0. In fact, otherwise we will have H 1 / 2 ( X l , X l ) H 1 / 2 ( x 2 , x2) ~ C d - ( N - 2 ) / 2

while G ( x 1 , x2) ~ Cd -(N-2).

Hence, for large d, C d - ( N - 2 ) / 2 ~ q~(Xl, x2) -- c• N - 2

which is impossible since c < 0. We observe that this conclusion does not depend on the value of c, but on the fact c is negative. By assumption, we also have IXl - x21 ~> 1. Then we let 6 --+ 0 and then assume that the point x2 converges to some 22 -- (2~, 2 ~ ) , where s u ~> 1. We also set, consis~ntly 21 -- (0, 1). The f u n c t i o n s / t ( x , y) and t~(x, y) converge to the corresponding ones H and G in the half-space X u > 0, namely to

A H ( x , y) =

bx

I x - ~ l N-2

and

( G ( x , y ) -- bN

,

1

)

Ix -- y [ N - 2 -- ix _ ~ I N - 2

"

Here, for y -- (yt, YN) we denote ~ -- (yt, - - Y N ) . Similarly, V~ converges to V~b, where q~(Xl, x2) - H 1 / 2 ( X l , X l ) H 1 / 2 ( x 2 , x2) - G ( x l , x2). We have that

~(~1, ~2) - 0 . -!

Assume first that x 2 ~ 0. Then A

Vx; ~(21,22) -- - V x ; G(21,22)

( --(N-2)bN

1 122-211 N-2

1 ), 122 - X ll N-2 x2 ~;~0

259

Bubbling in nonlinear elliptic problems near criticality

since clearly 13~2 - -~ll < la72 -- X 11. The vector of ]~N • R N (0 t, 0, X2, ' 0) is clearly tangent to the boundary of the restriction x u > 0, where we are assuming the considered point lies. Assume now that x 2' - - 0 , case in which otherwise x2 = (0', a0) with a0 >~ 2, and then

bNl~(.~l, .~2 -+- (a -- aO)Xl)

1

{

1

1

(2a)(U-a)/2 - ~ ( a -

2(N-2~/2

1) N - 2

(1 + a) N - 2

"

Differentiating with respect to a we get

bN1VxN ~(-~1,3C2)

-- __ (N __

- 2)[2 - ( N - l ) a0- N / 2

-- (a0 -- 1) - ( N - l )

+ (a0 + 1) - ( N - l )

].

This combined with the relation q3(Yl,-~2) --" 0 yields

bN17xX~(Xl,X2) = ( N -- 2 ) [ ( a 0

-

1) - ( N - l )

_ 2 - ( N - 1 ) a o N / 2 - (a0 -+- 1) - ( N - l ) ]

> 0.

-- 0 we have

Indeed, since ~b(s

1

,

1

2 N-laO/2 = 2a0 2N-2a(oN-2)/2 <

[1

1

( a 0 - 1)N-1 -- (a0--[- 1)N-1

1 "

So we can conclude that

bN 1VxN q~(Xl, X2) > 0.

D

In what follows, we prove the existence of a critical point for q-'2 when s is given by (3.1). In order to avoid the singularity of qz2 and q9 over the diagonal {~1 = ~2} we consider M > 0, we define

GM(~) =

{G(~)

M

i f G ( ~ ) <~ M, if G(~) > M,

(3.6)

and we replace G with GM in the definition of qz2 and qg. The choice of M will be made later on. Let r be fixed as in L e m m a 3.2. A direct consequence of L e m m a 3.2 is that, if 11 (M(~)) denotes the first eigenvalue of M (~), then

(~1, ~2) e o S N - 1 • o S N - 1 =ff l 1 (M(~)) < 0. Call S-oS

N-1 x o S N-1.

260

M. del Pino and M. Musso

For every ~ e S we let e(~) = (el (~), e2(~)) E R 2 be the negative direction of the quadratic form defined in (3.3). It is not restrictive to assume that le(~)! = 1. Furthermore, it is easy to see that there is a constant c > 0 such that c < el (~)e2(~) < c -1

(3.7)

for all ~ E S. Let Io -- [so, So 1] for some so > 0 and define bo--

max

~ E,_,C,SEI0

O2(~,se(~)),

ao--

max

~ES,sEOIo

q/2(~,se(~)).

Then one easily finds that, if so is small, then bo > ao. Define

w ~ ~ - {~ e s2 • s2. ~(~) < - ~ o } n (s2~ • s2~)

(3.8)

with q)o < min{ l e - 2 b ~ ' 1 2 max~e$ q)(~)} " It is easy to see that, for 6 small, S C W{ ~ Furthermore, we have that W~~ 1 7 6 {~ E ,f2 • ~f2" l l ( M ( ~ ) )

<-10}

(q (~f26 • S-23)

for an explicit l0 > 0. In particular, if we define

s2,

a = --~-to + log c -

1

+ 2 log so,

then ao < a. Let I -- {A = (A1, A2) E ~ 2 . following inequalities hold true bo ~>

min

A1A2 -- 1}. Then, taking so smaller if necessary, the

lP2(~,A) ~>

~ES,AEI

min

!P2(~,A) > a .

(3.9)

~EW~,AcI

Indeed, in (3.9) the first inequality follows if we take so < c, with c given by (3.7); the second inequality is trivial; the third inequality follows if we choose a < - M , which can always be obtained provided so is sufficiently small. THEOREM 3.2. There exists a critical level for q/2 between a and bo. PROOF. We first prove that for every sequence {(~n, An)} C W~ ~ • R 2 such that (~n, An) ---> (~, A--) ~ O(W~ ~ • R 2) and t//2(~n , A n ) E [a, bo] there is a vector T, tangent to O(W~ ~ x R2+) at (~, A), such that

v~2(~,x). T ~0.

0.~0>

Bubbling in nonlinear elliptic problems near criticality

261

In order to prove (3.10) we first observe that if An ~ A-E 0 ~ 2 then qJZ(~n, An) --+ - o c . Thus we can assume that X E R 2, ~ E ~ x ~ and ~0(~) ~< -(Po. Two cases arise, if VA 1I/2(~, A--) # 0 then T can be chosen parallel to VA T2 (~, A--). Otherwise, when VA qJ2 (~, A) -- 0 we have that A satisfies _

__

--2 A1

_

_

--

H(g2, g2) 1/2 H(~I, ~l)1/2(fi(~l, ~2)

and

H(~I, ~1) 1/2

A--22__

H(~2, ~2)l/2(fi(~l, g2)' and ~ satisfies (p(~) < 0. Substituting back in qJ2, we get 1 1 qJ2(~l, ~2, A-1 A-2) -- - ~ + ~ log

'

1

[(P(~I, ~2)1

and then ~o(~) <~ - e x p ( - 2 b o - 1) ~< -2(p0 < -(P0, so that necessarily ~ E 0 ( ~ x ~2~). At this point we choose M: we take 6 > 0 as in Lemma 3.3, then we let H~ = max{H(~l, ~1)/~1 E ff23} and consider M /> exp(2a - 1) + H~. We observe then that G(~I, ~2) ~< M. Thus, we can apply (3.5) to complete the proof of (3.10). Second we prove the Palais-Smale condition in W~ ~ x R 2 for qJ2 in the range [a, b0], that is, if {(~n, An)} C W~ ~ x R 2 satisfies qJ(~n, An) E [a, b0] and VtI/(~n, An) --->0 then {(~n, An)} has a subsequence converging to some (~, A--) E W~ ~ x R 2. In fact, it can be shown that the sequence An remains bounded. Then we conclude using (3.10). Assume now by contradiction that there are no critical value in the interval [a, bo] for T2. Because of what established above we can define an appropriate negative gradient flow that will remain in W~ ~ x ~ 2 in [a, bo]. Hence there exists a continuous deformation

such that for some a t < a, / ,

r/(0, u) = u

Vu ~ ~2 ~

r/(t, u ) - - u

Vu E (/t2 ,

a t

a f

r/(l, u) E (/-/2 . Let us call .A={(~,A)~W~ ~215

O.A= {(~,A) E W~ ~ • R 2 I~ E S , A - - r e ( ~ ) , r C = W~~ x I.

~ 010},

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M. del Pino and M. Musso

From (3.9) we deduce that r C ~:o, OA C tp~' and ~2a' A C - 0. Therefore n(o, u) - u

Vu 9 ,4,

O(t, u) - u

Vu 9 OA,

0(1, A) N C -

(3.11)

0.

For any (~, A) 9 A and for any t 9 [0, 1], we denote rl(t, (~, A)) = (~ (~, A, t), X(~, A, t)) 9 W[ ~ x R~. We define the set

13= {(~,A) E A I . 4 ( ~ , A , 1 ) E I }. Since rl(1, A) N C - 0 it holds 13 - 0. Now let/4 be a neighborhood of 13 in Wff~ • R 2 such that H* (/4) -- H* (/3). If yr" L/--~ S denotes the projection, arguing like in Lemma 7.1 of [33] we can show that yr*" H* (S) --+ H* (H)

is a monomorphism.

This condition provides a contradiction, since H* (b/) - {0} and H* (S) ~- {0}.

[]

Now we are in a position to complete the proof of Theorem 3.1, proving that the reduced functional J has a critical point. PROOF OF THEOREM 3.1 COMPLETED. Given the result of Lemma 2.2, we are left to

prove the existence of a critical point (~e, Ae) for J, for all e small enough. Equivalently, we are thus interested in critical points of J defined by J~(~, A)--OgN1 [J(~, A ) -

2CN

-- 8 y N ] .

We consider the domain Dr, R = W : ~ x [r, R] 2, with W[ ~ given by (3.8) and r, R to be chosen later. The functional J is well defined on Dr, R except on the set Ap -- {~ E Y2p x ~ p ] ]~1 - ~21 < P }. Taking into account Proposition 2.4 and proceeding as in (3.6) with the function G, we can extend J to all Dr, R, keeping the relations (3.5) and (3.9) over Dr, R. Arguing like in the proof of Theorem 3.2 and taking into account (2.74) and (2.75), there are positive numbers r, R, g and ~ such that, for all e 6 (0, g), for all ot 6 (0, &), the function J (~, A) satisfies the Palais-Smale condition in Dr, R in the range [a - c~, b0 + or]. Furthermore, if (~n, An) --+ (~, A--) E ODr,R with J~(~n, An) E [a - o r , bo + or] then there exists a vector T, tangent to Dr, R at (~, A) such that V J (~, A) 9 T r 0. Direct consequence of these two facts is that an appropriate negative gradient flow for J(~, A) can be defined and it remains in Dr, R in the range [a - o r , bo + or]. Using

Bubbling in nonlinear elliptic problems near criticality

263

again the Cl-closeness between ~P2 and f given by relations (2.74) and (2.75), one can prove estimates similar to (3.9) for J. Arguing as in the last part of the proof of Theorem 3.2, one shows the existence, for all e small enough, of a critical point ( ~ , A t ) (~le, ~2e, Ale, A2e) for J. Hence, the function 1

Us(X )

= E--2+e(N-2)/2 [g)~le,~;~ -q- U~.2e,~;~" "J- ~(~7;, A e ) ] @ - I / ( N - 2 ) X )

(see (2.6)) with )~i~ -- ( a N A 2 e ) 1 / ( N - 2 ) , is the solution to (2.1) we were looking for. In fact, one easily gets that this solution u~ has the following qualitative behavior

2 I )~je ] (N-2)/2 e -1/2 -ff o(1), Ue (X) -- Otn Z 2 2) ]2 j=l )~j~ -}- 8 - 2 / ( N - Ix -- ~J~ where o(1) is uniform over compact sets of F2.

D

Let us mention a problem where similar phenomena to those described take place: In [66], the authors studied problem (1.12) with # negative and small, say # --8 ( u - 4 ) / ( u - 2 ) for dimensions N ~> 5, namely /XU -- 8(N-4)/(N-2)U --~ U (N+2)/(N-2) --_ 0

in F2,

u>0

in~2, on0~,

u=O

(3.12)

with .(-2 a smooth bounded domain of ]~N, with N 7> 5. Despite of being critical and not supercritical, problem (3.12) shares some common patterns with (2.1). Indeed, if ~ is star shaped, Pohozaev's identity shows that (3.12) does not have any nontrivial solution. On the other hand, if G' is an annulus, then (3.12) has nontrivial positive solutions. In [66], a result similar to the one obtained in Theorem 3.1 for problem (2.1) holds true for (3.12). Indeed it is proven that in a general domain ~ with a sufficiently small hole then problem (3.12) has a positive solution u~ which, as e --+ 0, concentrates as the sum of two Dirac deltas around points ~1 and ~2 of the domain ~ . More precisely, for any 6 > 0, sup

u~(x) --+ O

Vj--l,2

xCB(~j,3)

and sup xEB(~j,(~)

ue(x) ~" 8 - ( N - 2 ) / ( 2 ( N - 4 ) )

Y j -- l, 2.

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M. del Pino and M. Musso

In this case, the location of the two points of concentration is defined through the critical points of the function A 1 q/2(~:l, ~2, A1, A2) -- ~ [H(~:I, ~:I)A 2 + H(~2, ~:2)A2 - 2G(~:l, ~2)A1A2] 1 + -~[A 4/(N-2) + A4/(N_2) ].

(3.13)

An argument similar to the one used to prove the existence of a nontrivial critical point situation for qJ2 given by in the slightly supercritical Bahri-Coron setting of Theorem 3.1 can be reproduced here, thus yielding to the following theorem. THEOREM 3.3. Assume that S2 - 79 \co,

(3.14)

where 79 and co are bounded domains with smooth boundary in ~N, N >>.5, with the property that co C B(O, p) C 79. There exists a Po > 0 such that, ifO < p < Po is fixed and s is given by (3.1), then there exists ~ > 0 such that, f o r any 0 < e < ~, problem (3.12) has a solution ue, which has the following f o r m

Ue(X) --

j=l

where o(1) is in C~

()~j)2 -Jr-6-2/(N-4)Ix -- ~jI 2

(N-2)/2] e-(N-2)/(2(N-4)) -Jr-o(1),

over compact sets o f s

as e--+ O. Here (I~)N-2 =

yEN U2 (fEN ~p)2 (A~) 2 and ( ~ , ~ , Ael , A~) converges (up to subsequences), as e --+ O, to a critiA

cal point o f the function q/2 defined in (3.13).

The result obtained in Theorem 3.1 (and also in Theorem 3.3) is consequence of the following more general theorem, which we state for the slightly supercritical problem (2.1) and which has been proved in details in [33]. THEOREM 3.4. Let s be a bounded domain with smooth boundary in ~N, with the following property: There exist a compact manifold .A4 C s and an integer d >>,1 such that q9 < 0 on .All x ./k4, t* : H d (s --+ H cl(.All) is nontrivial and either d is odd or H 2d (s = O. Here q9 is the function defined in (3.4). Then there exists e0 > 0 such that, f o r any 0 < e < eo, problem (2.1) has at least one solution uE. Moreover, let C be the component o f the set where 99 < 0 which contains .All • ~A/[. Then, given any sequence e -- en --+ O, there is a subsequence, which we denote in the same way, and a critical point (~1, ~2) ~ C o f the function 99 such that u~ (x) --+ 0 on compact subsets o f S-2 \ {~1, ~2} and such that, f o r any 8 > O,

sup Ix-~il<8

u~(x)--+ +cx~,

i - - 1,2,

Bubbling in nonlinear elliptic problems near criticality

265

as e -+ O.

In view of Theorem 3.4, observe that the perforation in Coron's situation contained in Theorem 3.1 need not to be symmetric or contained in a small ball. For instance in R 3, the result holds true in a domain with a torus with narrow section excised. A natural question now concerns the issue of solvability of problem (2.1) in a domain exhibiting multiple holes. Next result asserts that in such a situation, multipeak solutions exist, consisting of gluing of double-spikes associated to each of the holes. More precisely, we have the theorem. THEOREM 3.5. Let 77 be a bounded, smooth domain in R N, N >~ 3, and P1 . . . . . Pm points o f 77. Let us consider the domain m

(3.~5) j=l where p > O. There exists a Po > O, which depends on 77 and the points P1 . . . . . Pm such that if 0 < P < Po is fixed and 1-2 is the domain given by (3.15), then the following holds: Given a number 1 <~ k <, m, there exists ~ > O a n d a family o f solutions, ue, 0 < e < ~ o f (2.1), with the following properties: u~ has exactly k pairs o f local maximum points (~.~1, ~j2 ) E S'22, j -- 1 . . . . . k, with cp < I~i - Pjl <, Cp f o r certain constants c, C independent o f p, and such that, f o r each small ~ > O,

sup

u e (x) ~ 0

{Ix-~jil>8 vi,j}

and

sup

ue (x) --+ +(x~

Vi, j

{Ix-~ji 1<~} as e --+ O.

The result contained in the previous theorem is proved in detail in [32]. We will skip the proof of it and we refer the reader to [32]. We just want to mention few facts here. Thanks to the fact that the holes in (3.15) are spherical, we can be extremely precise in the location of each couple of peaks: indeed, as explained in the theorem, they are located at distance of order p from the hole. Not only that. From the proof it follows that they are antipodal with respect to the hole. On the other hand, from the proof of this result it is clear that, in order to get the result on existence of multispike solutions, there is no need for the small excised domains to be balls of the same radii. We consider only this case for notational simplicity. Let us also observe that by re-labeling the points P1 . . . . . Pm, the above result actually yields that, for each 1 ~< k ~< m and any set of indices il . . . . . ik in {1. . . . . m }, a solution exhibiting double-spikes simultaneously near the points Pi~ . . . . . Pik exists. This in particular

266

M. del Pino and M. Musso

yields the existence of at least 2 m - 1 solutions of the problem whenever e is sufficiently small.

3.2. Bubbling under symmetries As it was natural to expect, Theorem 3.5 shows that multiplicity of solution to (2.1) appears in presence of multiple holes. What we want to show next is that the concentration phenomena involved in problem (2.1) when e is sufficiently small, is in fact much richer than it might be a priori suspected, even in domains with just one hole, at least in the case of domains exhibiting symmetries. In particular, we find the presence of a large number of geometrically distinct solutions to problem (2.1) when S2 is an annulus,

Aba--{xla
(3.16)

for given 0 < a < b, provided that e > 0 is sufficiently small. More precisely, we find that a k-spike solution of (2.1) exists for any k sufficiently large. The same result holds true in the solid torus Tab, 0 < a < b in IR3, obtained after rotating around the x3-axis a disk of radius (b - a)/2 centered at the point ((a 4- b)/2, 0, 0). Our main result contained in Theorem 3.6 covers these examples, and also the case of a solid of revolution, symmetric on the variable x3, which does not contain the origin. See also [23,49,61 ] for related problems. For notational simplicity, we write x E ]I~u as x -- (Z, x t) with z 6 ]1~2 which we identify with an element of the complex plane, and x I E •N-2. THEOREM 3.6. Assume that the domain 1-2 in the following properties: (a) For any z, x',

(z, x') E ~

implies

]~N, N

~ 3, is such that 0 ~ 12 and satisfies

(e i~ x') E $2 for all 0 E [0, 2rt].

(b) For each 3 <, i <, N, (Z, X3 . . . . .

Xi . . . . .

XN) E

F2

implies

(z, X3 . . . . . --Xi . . . . . X N ) E a"-2.

For k E N, let us denote Pj - - (e 2rtij/k, Or), j -- 1 , . . . , k. Then there is a ko, such that for k ~ ko there exists a positive number eo such that for each 0 < e < eo there are numbers )~E and pe and a solution ue to problem (2.1) of the form

U e ( X ) ~-" OIN jZ= l

)~2 + E - Z / ( N - 2 ) [ x __ p e P j l 2

E -1/2 4- o(1),

(3.17)

where o(1) ~ 0 uniformly as e --+ 0 and OIN - - ( N ( N - 2)) (N-2)/4. Moreover, )~ is bounded above and below away from zero and pe Pj remains uniformly away from the boundary of F2.

Bubbling in nonlinear elliptic problems near criticality

267

As it will become clear from the construction, the number k may in principle need to be chosen very large for a k-peak solution to exist. According to the results of Lemma 2.2 in the previous section, our task reduces to finding critical points of the functional of (~, A) given by J(~, A) = Ie(V + 4)). Let (a, b) be any maximal interval so that p P1 E 12 for all p 6 (a, b). The setting of Theorem 3.6 suggests us to restrict ourselves to seek for critical points of the form

~j--pPj,

Aj =/.t,

forall j -

1 . . . . ,k,

(3.18)

where p E (a, b) and tt > 0. Let us set

J~(P, #) = J(P(P1 . . . . . Pk), #(1 . . . . . 1)).

(3.19)

LEMMA 3.4. Under the assumptions of Theorem 3.6, if (p, tt) is a critical point of J, then (~, A) -- (P(P1 . . . . . Pk), #(1 . . . . ,1)) is a criticalpoint of J. PROOF. We first investigate the symmetries inherited by the function r k) in the variables (Yl . . . . . YN) when the points (~, k) belong to 12. We recall that r satisfies p+e

A ( V + 4)) + (V + 4))+

1

= Y-~ij cij Vip- Zij

0--0

fs?~ 4) Vip- 1Zij -- 0

in 12e, on 012~, for all i, j.

(3.20)

Now, because of the symmetry of the data and the domain, we see that V and each of the Vi's (see (2.7)) are even with respect to each of the variables Y2. . . . . YN. It then follows that Cij - - 0 for all 1 ~< i ~ k and for all 2 ~< j ~< N. In fact, for instance for j = 3, the function ~b(yl, Y2,-y3 . . . . . YN) satisfies an equation of the same form, with (possibly) different constants C i j . Uniqueness of the solution r to the above problem yields that 4) is even with respect to Y3. For i = 1 then only the terms Cll V~ -1Zll 4- cl(U+l) V~ -1ZI(N+~) are present in the summation on the fight-hand side of (3.20). On the other hand, the invariance under rotations assumed implies that actually 0 is symmetric with respect to the line z = te 2rril/k in the space of the first two variables. Call 2~1

ZI - Zll cos ~

k

4- Z12 sin

2~rl k '

namely the HI-projection of the directional derivative of ~ in the direction of the above line. Then by the symmetry, we obtain that the fight-hand side of (3.20) reduces simply to an expression of the form y-~4Cl vtP-Iz1 + czZI,N+I, where ci - c i ( p , It), i - 1,2. Thus according to the previous section, (~, A) = (P(P1 . . . . . Pk), #(1 . . . . . 1)) is a critical point of J provided that ci(p, 1s ~ 0 for i --,~1, 2. We claim that these relations hold true if (p, #) is a critical point of J. In fact, V J - 0 means

DI~(V + O)

(U + O) = O = D I ~ ( V + O )

- ~ ( V + O) 9

(3.21)

268

M. del Pino and M. Musso

Now, k 0 (V Op

+

ok) _

k 0 ( V + ~ ) -- ~ 0/z

Z Zl + o(1), /=1

ZI(N+I) + o(1)

/=1

with o(1) --+ 0 uniformly as e ~ 0. So we have that relations (3.21) are respectively equivalent to

(1 - o ( 1 ) ) C l + o ( 1 ) c 2 = 0,

(1 - o(1))c2 + o(1)Cl = 0,

hence r -- C2 ---- 0, and the proof is concluded.

U

PROOF OF THEOREM 3.6. From Lemma 3.4, we need to find a critical point of the functional J (/9,/z). This is equivalent to finding a critical point of t//~ ( p , / Z ) ~ (_ON1 (6-1 ( J " ( p , / z ) -- k C N ) - YN)

(3.22)

for the constants introduced in expansion given by Proposition 2.4. Now, again from Proposition 2.4, !Pe (p,/z) = q~0(p,/z) + o(1), where o(1) is small in e, in the C 1-sense, uniformly on compact subsets of (a, b) x (0, oo), and

tPo(P, lZ)=qJ(p(P1 . . . . , Pk),/z(1 . . . . . 1)) = k -~-Fk(p) + l o g / z

(3.23)

with k F k ( p ) -- H ( ~ I , ~1) - Z G(~I, ~i) i=2

and ~j = I9 P j . Since Robin's function H (~, ~) tends to + ~ as ~ approaches 012, it follows that for any integer k, limp~a Fk(p) = limp~b Fk(p) = +oc. On the other hand, if ~j = a+__Ob pj, then G(~I, ~2) - YN[~I -- ~212-N @ O(1),

where the quantity O(1) is bounded independently of k, hence G(~I, ~2)/> AkN-2 for all large k, with A independent of k. Now, H (~1, ~1) ~< B with B independent of k. It follows that

Fk

a+b) 2 <~k(B - Ak N-2) < 0

Bubbling in nonlinear elliptic problems near criticality

269

for all sufficiently large k. Then there are numbers a < h 0 and Fk (p) < 0 hold for all p 9 (fit/~). Then if 6 is fixed and sufficiently small we see that the following relations hold O q,'o(p, ~) > o, o~

3 qJo(P ~-1) < 0 a~

for all p 9 [fi b]

(3.24)

0 opOO(fi, #) < O,

0 OpOO(/~#) > 0

for all # 9 [~ ~ 1].

(3.25)

and

Let us set 7g = (fi, b) x (3, 3 -1) and let (dl, d2) be the center point of this rectangle9 Let us consider the homotopy Ht(p,/~) -- tVtPo(p,/~)

-+- (1 - t ) ( p - d l , - ( / r

- d2)),

t 9 [0, 1].

Then, from relations (3.24) and (3.25), we see that the degree deg(Ht, 7-4, 0) is well defined and constant for t 9 [0, 1]. It follows then that deg(VqJ0, 7"4,0) = - 1 . Since Vq/e is a small uniform perturbation of VO0 on 7g, we conclude that deg(VqJe, 7g, 0) -- - 1 for all sufficiently small e. Hence a critical point (pe,/ze) 9 7r of q/e indeed exists for all sufficiently small e and the proof of the theorem is thus concluded. 89 REMARK 3.1. In the limit (pe,/zc) --+ (p,/z) as e --+ 0 we find a critical point of the function q/0 in (3.23). Computing directly, we see that #2=_

1 (3.26)

kFk(p) '

relation that makes sense only if Fk (p) < 0. Moreover, a refinement of the region 7g in the above proof yields that p can actually be chosen to be a minimum of Fk on (a, b).

4. The Brezis-Nirenberg problem in dimension N = 3: the proof of Theorem 1.2

In this section we construct single and multiple-bubbling solutions for problem (1.13) in dimension N = 3. Unlike the case of higher dimensions, the results of [15] concerning asymptotic analysis of radial solutions in a ball when the exponent approaches critical from below, suggests that the object ruling the location of blowing up in single-bubble solutions of (2.1) is Robin's f u n c t i o n gx defined as follows. Let )~ < X1 and consider Green's function G)~(x, y), solution for any given x 9 I2 of -AyGx

- )~Gx = 6x,

G x ( x , y) = O,

y 9 I2, y 9 0s

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M. del Pino and M. Musso

Let Hz (x, y) = F (y - x) - Gz (x, y) with F (z) = 4@tzl' be its regular part. In other words, Hz (x, y) can be defined as the unique solution of the problem Ayn)~ + XH~. - X F ( x -- y),

yEs

Hz = F ( x -

y E Os

y),

Let us consider Robin's function of G)~, defined as g)~(x) = H)~(x, x). It turns out that gz (x) is a smooth function which goes to +cx~ as x approaches 0s Its minimum value is not necessarily positive. In fact this number is strictly decreasing in X. It is strictly positive when X is close to 0 and approaches -cx~ as X ]' Xl. It is suggested in [15] and proven by Druet in [38] that the number X* given by (1.4) can be characterized as

X * = sup{X> 0: mingz > 0].

(4.1)

s

The minimum value of gz is thus negative whenever X, < X < Xl. In what follows we shall denote simply by 79 the subset of s where gz is negative and q = 5 + e. Further we shall present general scheme of the proof. Full details can be found in [30].

4.1. Energy expansion of single bubbling Given a point ~" E R 3 and a positive number #, we denote in what follows 31/4

wu,~" (Y) =

/z-l/2 v/1 +/Z - 2 lY - ~"I2

which correspond to all positive solutions of the problem Aw+w 5=0

i n n 3.

The solutions we look for in Theorem 1.2, part (a) have the form u(y) ~ wu,r (y), where E s and # is a very small number. It is natural to correct this initial approximation by a term that provides Dirichlet boundary conditions. We assume in all what follows that 0 < X < X1. We define zru,r (y) to be the unique solution of the problem Ajr#,~. + Xzru, ~- = -Xwu, ~- in s

with (4.2)

7r#,~- = - - w # , ~ -

on 0s

Fix a small positive number # and a point ~" E s the solution one of the form Uu, ~-(y) = wu,~ + 7ru,~-.

We consider as a first approximation of

(4.3)

Bubbling in nonlinear elliptic problems near criticality

271

Observe that U = Uu,r satisfies then the equation AU + X U + w ,5, ; - - 0

int2

U=O

on 3t2.

Classical solutions to (1.13) correspond to critical points of the energy functional

Eq,)~ (u) ~ -~l f s ? IOu 12 - -~ ~'fs2 lu 12

1

q+l

fs?

lu

Iq+l

9

(4.4)

Here and in what follows we denote q = 5 + s. If there was a solution very close to Uu.,;. for a certain pair (#*, ~'*), then we would formally expect Eq,)~ to be nearly stationary with respect to variations of (#, ~') on U , , ; around this point. Under this intuitive basis, it seems important to understand critical points of the functional (#, ~') w-~ Eq,x (Ulz, ~). We have the following explicit asymptotic expression for this functional. For q close to 5. LEMMA 4.1. Consider U~,, and Eq,z defined respectively by (4.3) and (4.4). Then, as lz --+ O, Eq,z(Uu,~) -- ao + al/zgx(~') + a2/z2X - a3/z2g2(~ ")

+ (q - 5)[a4 log # + as] + (q - 5)201 (if,/Z, q) + / z 3-or 02(if,/Z, q),

(4.5)

where f o r j = 0 , 1,2, i = 0 , 1, i + j ~ 2, l = l, 2, oi+J

lz j

Off i olzJ

Ol(~, #, q)

is bounded uniformly on all small lz, s = q - 5 small and ~ in compact subsets of 12. Here ao . . . . . a5 are explicit constants.

Let us consider then q = 5 + s and assume that ~" 6 79, i.e., gx (~') > 0. It is convenient to consider A defined by a4 1 /z ---- - s - - ~ A, al g)~(~')

(4.6)

where a4 and a l are the constants in the expansion (4.5). LEMMA 4.2. In the situation of Theorem 1.2, part (a), f o r # given by (4.6), consider a functional of the form ~e(A, ~') = Es+s,, (U,,~-) -+- eOe(A, ~)

M. del Pino and M. Musso

272

for A > 0 and ( ~ Y2. Denote V = (OA, 0~ ) and assume that

10~1-4-IV0~l + [VOAOEI ~ 0

(4.7)

uniformly on ((, A) as e --+ O, with 6 < A < 6-1

gz(() < -3

for any given 6. Then ~ has a critical point (Ae, (e) with ~e ~ 79, Ae --+ 1,

g~. ((~) --+ min g~. $2

PROOF. The expansion given in Lemma 4.1 implies

ao+a4[Z+,og+,og( )] 05] + e O e ( A , ( ) , +a4e [ log (a_14) + l o g e + m a4

where 0e still satisfies (4.7). The main term in the above expansion is the functional ~o(A,f)---A+logA+log(

1 ) gz(() '

which obviously has a critical point since it has a nondegenerate maximum in A at A = 1. Consider the equation OA~e(A,()--O,

which has the form A-

1 4-o(1)0e(A, (),

where the function 0E has a continuous, uniformly bounded derivative in (A, () in the considered region. It then follows that for each ( 6 79 there exists a unique A -- Ae((), function of class C 1 satisfying the above equation which has the form Ae(()-

1 + o(1)fle((),

where fl~ and its derivative are uniformly bounded in the considered region. Clearly we get a critical point of Oe if we have one of the functional ( w-~ 7tc(Ae ((), (). Observe that on 79

E (1)

7ts(As((), () -- cs + a4s log

g~(()

+ o(1)

]

Bubbling in nonlinear elliptic problems near criticality

273

where o(e) is small uniformly on 79 in the Cl-sense and ce is a constant. The linking structure is thus preserved, and a critical point ~'e ~ 79 of the above functional with the desired properties thus exists. This concludes the proof. D

4.2. The method of proof Our purpose in what follows is to find in each of the situations stated in the theorems, solutions with single or multiple bubbling for some well chosen ~" ~ Y2, which at main order look like k

(4.8)

U -~ E (to#i,~" "3I-Y'Y#i,~") i=1

with #1 small and, in case k >~ 1, also with/zi+ 1 ((/Z i . This requires the understanding of the linearization of the equation around this initial approximation. It is convenient and natural, especially in what concerns multiple bubbling to recast the problem using spherical coordinates around the point ~ and a transformation which takes into account the natural dilation invariance of the equation at the critical exponent. This transformation is a variation of the so-called Emden-Fowler transformation, see [44]. Let ~" be a point in s We consider spherical coordinates y -- y(p, 0 ) centered at ~" given by

y-~ P=IY-~'[

and

69=

l y -~ l '

and the transformation T defined by

v(x, O))--7-(u)(x, O))=--21/2e-Xu(~ -+- e-2X69).

(4.9)

Denote by D the ~'-dependent subset of S -- IR x S 2 where the variables (x, 69) vary. After these changes of variables, problem (2.1) becomes

{

4As2v-k-vtl--v+4~,e-4Xv+cqe(q-5)xv q = 0 v>0 v--0

with Cq -~

2 -(q-5)/2.

Here and in what follows, " ' " -

a__ We observe then that

T(w.,~)(x, o ) = W(x -~),

inD, inD, on OD

(4.10)

274

M. del Pino and M. Musso

where

W(x) =-- (12)l/4e-X (1 -+-e-4X) -1/2 -- 31/4[cosh(2x)] -1/2 and # - e -2~ . The function W is the unique solution of the problem W"-

W + W5 -0

on ( - o o , oo),

w'(o) -0, W>0,

W(x)--+O

asx~+~.

We see also that setting H~,C = T(zru, c)

with/z - - e -2~,

t h e n / 7 = H~,C solves the boundary value problem - (4 A S217 ql._ / 7 ff __ / 7 -11- 4Xe - 4 x / 7 ) = 4Xe -4x W (x - ,})

in D,

I-1- -W(x

on OD.

- ~)

An observation useful to fix ideas is that this transformation leaves the energy functional associated invariant. In fact associated to (4.10) is the energy

Jq,x(v) = 2 fD IVovl 2 -Jr-~l f D [Iv t 12 -Jr-11312]

__ 2X fD e_4Xv2

Cq q+l

fDe(q_5)Xlvlq+l.

(4.11)

If v = 7-(u) we have the identity

4Eq,z (u) -- Jq,z (v). Let ~" 6 S2 and consider the numbers 0

< ~1 < ~2 < "'" < ~k. Set k

Wi (x) : W (x - ~i),

Hi

- - 1"I~i,~,

~/~ - - W i .qt_ Hi

,

V= ~ V / . i=1

We observe then that V - T ( U ) where U is given by (4.8) and/Z i - - e -2~i . Thus finding a solution of (2.1) which is a small perturbation of U is equivalent to finding a solution of (4.10) of the form v = V + r where r is small in some appropriate sense. Then solving (4.10) is equivalent to finding r such that

{

L(r r

=-N(r

onOD,

- R,

Bubbling in nonlinear elliptic problems near criticality

275

where L(q~) = 4As2q~ + 4)" - 4) + 4)~e-4X~b + q c q e (q-5)x V q-1 ~,

(4.12)

N(dp) =~ c q e ( q - 5 ) x [ ( v + dp)q+ - V q - q v q - l d p ]

(4.13)

and k

R =-- Cqe (q-5)x V q - Z

(4.14)

We.

i=1

Rather than solving (4.10) directly, we consider first the following intermediate problem: Given points $ = ($1 . . . . . ~k) 6 R k and a point ~" 6 s find a function 4~ such that for certain constants Cij, [ L(~b) -- - N ( d p ) - R + Z i , j c i j Z i j 4~--0 fD Zij 4~dx dO - 0

in D, on OD, for all i, j,

(4.15)

where the Zij span an "approximate kernel" for L. They are defined as follows. Let Zij be given by Zij(X, (~) -- 'T(Zij), i = 1 . . . . . k, j = 1 . . . . . 4, where zij are respectively given by 0 z i j ( Y ) - -Z--y-~w m , ~ ( y ) , ogj

j - 1 . . . . . 3,

0 Zi4(Y) - - / z i q---L--//)#i,ff (Y), OlZi

i -- 1 . . . . . k,

with/z i e -2~i . We recall that for each i, the functions zij for j = 1 . . . . . 4, span the space of all bounded solutions of the linearized problem =

AS -+- 5//) 4m,cZ - - 0

in R 3 .

This implies that the Zij'S satisfy 4 A s 2 z i j + Zi~ -- Zij + 5 W ? z i j --O.

Explicitly, we find that setting Z ( x ) -- 121/4e -3x (1 -+- e-4X) -3/2 -- 3 1 / 4 2 - 1 [ c o s h ( 2 x ) ] -3/2,

we get Zij '- Z ( x -- ~ i ) O j ,

j = 1, 2, 3,

Zi4 = W ' (X -- ~i).

276

M. del Pino and M. Musso

Observe that f~•

2 ZijZil -- 0

for 1 :/: j.

The Zij a r e corrections of Zij which vanish for very large x. Let OM(S) be a smooth cut-off function with

17M(S) --0

for s < M,

rIM(S)= 1 for s > M + 1.

We define

Zij -- (1 - ~TM(X - ~ i ) ) z i j , where M > 0 is a large fixed number. We will see that with these definitions, problem (4.15) is uniquely solvable if the points ~i, ~" satisfy appropriate constrains and q is close enough to 5. After this is done, the remaining task is to adjust the parameters ~" and ~i in such a way that all constants Cij --- O. We will see that this is indeed possible under the different assumptions of the theorems.

4.3. The linear problem In order to solve problem (4.15) it is necessary to understand first its linear part. Given a function h we consider the problem of finding 4~ such that for certain real numbers r the following is satisfied { L(4~) -- h + Y~i,j cijZij 4)=0 fD Zij~ --0

in D, on 0D, for all i, j.

(4.16)

Recall that L defined by (4.12) takes the expression L(~b) -- 4As2~b + 4)" - 4) + 4,ke -4x 4) + qcqe (q-5)x V q-1 ~. We need uniformly bounded solvability in proper functional spaces for problem (4.16), for a proper range of the ~i's and ~'. To this end, it is convenient to introduce the following norm. Given an arbitrarily small but fixed number a > 0, we define

k Ilfll, sup co(x) - l l f ( x , (9)1 with og(x) = Z e - ( 1 - ~ ) l x - ~ i l (x,O)cD i--1 m

We shall denote by C. the set of continuous functions f on D such that

IIf II, is finite.

Bubbling in nonlinear elliptic problems near criticality

277

PROPOSITION 4.1. Fix a small number 3 > 0 and take the cut-off parameter M > 0 of Section 4.2 large enough. Then there exist positive numbers eo, 6o, Ro and a constant C > 0 such that if Iq - 51 < eo, 0 ~ /~ ~ ~.1 -- 3,

and the numbers 0

dist(~', 0s

< ~1 < ~2 < "'" < ~k

Ro < ~1,

(4.17)

> 60,

satisfy

Ro < min (~i+1 - ~ i )

(4.18)

l<.i
with ~k < ~ ~o if q r 5, then f o r a n y h ~ Ca(D) with Ilhll, < +co, problem (4.16)admits a unique solution 4) ---- T (h). Besides,

IIT(h)[I, ~< CIIhll,

and

Icijl <~ CIIhll,.

4.4. Solving the nonlinear problem In this section we will solve problem (4.15). We assume that the conditions in Proposition 4.1 hold. We have the following result. LEMMA 4.3. Underthe assumptions of Proposition 4.1 there exist numbers co > 0, C] > 0 such that if ~ and ~ are additionally such that IIRll, < co, then problem (4.15) has a unique solution r which satisfies

114,11,~< C1 IIRII,. PROOF. In terms of the operator T defined in Proposition 4.1, problem (4.15) becomes r where N(r region

T(N(r

-Jr-R) = A(r

(4.19)

and R where defined in (4.13) and (4.14). For a given R, let us consider the

for some ?, > 0, to be fixed later. From Proposition 4.1, we get

IIA<, I[, Co[llN<,)ll, + IIRII,]. On the other hand, we can represent

N(r

= cqe(q-5)X q(q - 1) fo 1(1 - t) dt [V

+ tr162

2,

278

M. del Pino and M. Musso

so that (making q - 5 smaller if necessary) IN(~)I ~ Cllq~l 2, and hence IIN(r C1114~1]2,. It is also easily checked that N satisfies, for 4~1, r E S'• IIN(~l) - N(gb2) II, ~< C2y IIRII,II~ - ~211,. Hence for a constant C3 depending on Co, C1, C2, w e get

IIA
4~211,.

With the choices y = 2C3,

IIR II, ~ co =

4C~'

we get that A is a contraction mapping of,T'• and therefore a unique fixed point of A exists in this region. This concludes the proof. D After problem (4.15) has been solved, we will find solutions to the full problem (4.10) if we manage to adjust the pair (~, ~') in such a way that cij (~, ~') = 0 for all i, j. This is the reduced problem. A nice feature of this system of equations is that it turns out to be equivalent to finding critical points of a functional of the pair (~, ~') which is close, in appropriate sense, to the energy of the single or multiple bubble V. We make this precise in the next section for the case of single bubbling, k = 1.

4.5. Variational formulation of the reduced problem for k = 1 Next we assume k = 1 in problem (4.15). We omit the subscript i = 1 in Cij , Z i j and ~i. Then in order to obtain a solution of (4.10) we need to solve the system of equations

cj (~, ~') -- 0

for all j - 1 . . . . . 4.

(4.20)

If (4.20) holds, then v = V + 4~ will be a solution to (4.10). This system turns out to be equivalent to a variational problem, as we discuss next. Let us consider the functional Jq,z in (4.11), the energy associated to problem (4.10). Let us define

F(lz, ~) =-- Jq,z(V +~b),

/ z - e -2~,

(4.21)

where 4~ = 4)(~, ~') is the solution of problem (4.15) given by Proposition 4.1. Critical points of F correspond to solutions of (4.20) under a mild assumption that will be satisfied in the proofs of the theorems, as we shall see further.

Bubbling in nonlinear elliptic problems near criticality

279

LEMMA 4.4. Under the assumptions of Proposition 4.1, the functional F ( ( , ~) is of class C 1. Assume additionally that R in (4.14) satisfies that IIRll, ~< ~8~ where cr > 0

is the number in the definition of the ,-norm. Then f o r all lZ > 0 sufficiently small, if V F (~, () = 0 then (~, () satisfies system (4.20). We have now all the elements for the proof of our main results regarding single bubbling.

4.6. Proof of Theorem 1.2, part (a): single bubbling We choose # as in (4.6), a4

1

#=-e--~A, al g z ( ( ) where e = q - 5. We have to find a critical point of the functional F (/z, () in (4.21) for q = 5 + e. Consider

R -- cqe (q-5)x ( W ( x - ~) -+- 17~(x, 69)) 5+e - W ( x - ~)5, where e -2~ --/z. We write as usual W1 - W (x - ~), V -- Wl + H~. Then we can decompose R = R1 + R2 + R3 + R4, where R 1 ~

e ex (V5+ e _ VS),

R3 = V 5 - W 5,

R2 ~ V 5 (e e x - 1), R4 = (Cq - 1)e ex V 5+e.

We have

R1 ~ ee ex fo 1(1 - t ) d t (V 5+te log V), from where it follows that IRI[ ~< Ceee~eelx-~lv 4+1/2 ~ CEV 4. Since I/-/~1 ~ C e-(x+~) <. C e - l x - ~ l , we get IRll ~ C e e -41x-~l and hence IIRIII, ~ Ce. Direct differentiation of the above expression, using the bounds for derivatives of H~ yields as well

[10 2R II, +

]10~R1 [[, + IIO~Rlll, ~ E.

Let us denote V = [0~, 0~ ]. Thus we have

IIRIlI, + IIVR~ II, + IIVO~R1 II, ~ C~.

280

M. del P i n o a n d M. M u s s o

Observe that the same estimate is also valid for R4. On the other hand, we have

R2 -- g 5

(eex - 1) = sx V 5 fo 1 e tsx dt.

Since ~ ~ c log(1/s), we obtain for R2 and derivatives the bounds liB211, + IlVRz[I, + IlV0~Rzll, ~ Gel logel. Finally, for

R 3 - - 5 f o 1 (1 -- t)dt (Wl -]- t//~)4/7~, we find the bound IR31 ~ C e - ~ - x - 4 1 x - ~ l ~ Ce -2~-Ix-~l, and similarly for derivatives. We get, recalling that e -2~ = / z ~< Cs, lie3 II, + live3 II, + IIV0~R311, ~ Ce -2~. Concerning R4, a direct computation gives IR41 ~< Cse -51x-~l . Thus for full R we have [IR I[, + 1[V R [[, + 11VO~R 11, <~ Cs[ log el. It follows from Lemma 5.20 that for this choice of/z,

F(~, () -- Jq,z(V) +/z2l log/zl20(~, () 1 with ]0] + [VO~0] + IV0] ~< C. Define ~ps(A, () = F(89log ~, () with/z as above. A critical point for ~ps is in correspondence with one of F. We conclude that

~s(A, () -4Es+s,~(U~,~) + sOs(A, () with 0s as in Lemma 4.2. The lemma thus applies to predict a critical point of ~Ps and the proof of part (a) is complete.

4.7. Proof of Theorem 1.2, part (b): multiple bubbling Let us consider the solution 4~(~, () of (4.15) given by Proposition 4.1 where ~ = (~1, ~2 . . . . . ~k). Choosing ( - 0 makes 4) symmetric in the (~9i variables, which automatically yields cij 0 for all i -- 1 . . . . . k and j = 1, 2, 3. Thus we just need to adjust ~ in such a way that r - - 0 for i -- 1. . . . . k. Arguing exactly as in the proof of Lemma 4.4 we -

-

Bubbling in nonlinear elliptic problems near criticality

get that this is equivalent to finding a critical point of the functional F(r where ~" has been fixed to be zero. Similarly as before, we find now that

Jq,)~(W) -+- (llell 2 +

f(~)-

281

= Jq,~ (V + ok),

IIO~ell2)O(~),

where 0 and its first derivative are continuous and uniformly bounded in large ~. In what remains of this section we fix a number 6 > 0, set e = q - 5 > 0 and choose # i - - e - 2 ~ i in order that ~1 --

8A1,

#j+l

# j ( A j + 1 8 ) 2,

--

j -- 1 , . . . , k - 1,

(4.22)

with

< A j < ~-1,

j--1,...,k.

(4.23)

Let us measure the size of 11RI[, and 1[0~ R [1, for this ansatz. We can now decompose R = R1 + R2 + R3 + R4 + R5 where R1 = C x (V 5+e - Vs),

R3 = V 5 -

Wi

R2 = V 5 (C x - 1),

,

R4 ~

Wi

-

W5,

R5 -- (Cq - 1)C x V 5+C. We can estimate k-1

[R41 ~< C Z

e - ( ~ i + l -~i ) e - 3 1 x - ~ i t

i=1

hence IIR4II, ~< C e, a similar bound being valid for its derivatives in ~i's. The quantities R j for j -- 1, 2, 3, 5 can be estimated in exactly the same way as in the proof of Theorem 1.2, part (a). Thus IIRII, + II0~RII, ~< C e l l o g e l . Let us set A = (A1 . . . . . Ak) and define ~c(A) = F(~) with ~ given by (4.22). We need to find a critical point of ~e. We have proven that

lOe(A) -- Jq,z(V) + O(e21 log el2)0e(A),

(4.24)

where 0e and its first derivative are uniformly bounded. We have the validity of the following fact, whose proof we postpone for the moment 1 1 -~Jq,)~(V) - kao + [ ~ , ( A ) + o(1)]e + -~k(k + 1)a4ellogel,

where k

llr,(A) -- algz(O)A1 + ka410gA1 + E [ ( k j=2

j + 1)a410gAj - a6Aj]

(4.25)

282

M. del Pino and M. Musso

and the term o(1) as e ~ 0 is uniformly small in C 1-sense on parameters A j satisfying (4.23). Here the constants a0, a l, a4 are the same as those in Lemma 4.1, while a6 -- 16rt4c3. The assumption gz (0) < 0 implies the existence of a unique critical point A, which can easily be solved explicitly. It follows that o(1) C 1 perturbation of ~p, will have a critical point located at o(1) distance of A,. After this observation, the combination of relations (4.25) and (4.24) give the existence of a critical point of Oe close to A, which translates exactly as the result of Theorem 1.2, part (a). It only remains to establish the validity of expansion (4.25). We recall that k

k

i=1

i=1

where U = ~/k_ 1 Wi "[- 7ri, and we denote Wi -- W#i,O, Yri - - 2T#i,O, Ui = that Jq,z (V) -- 4Eq,z (U), where q = 5 + e. Observe that we can write

ttJi + 7ri.

We have

Eq,)~ (U) -- Es,z (U) + ~ , where (e (q-5)x -- 1)IV[ 6 + 4rtAq. A direct computation yields A q - - k ( q - 5)

( ~1 f c~ 1 FW6dx)+~ oc W 6 log W dx + ~-~

On the other hand, T'g - 4 rt A q = --61 fD [e(q-5)x -- 1 ] V 6 d x

1

- ~ ( q - 5)4rt

f x v6dx + o(e)

- 2 ( q - 5)

~i

dx + 0(8)

i=1

oo

k

-- a4 (q -- 5) Z

log/zj + o(e).

j=l

Now we have

g5,~. (U) = Z g5,~. ( U j ) -~j=l

Vi 6 -

i=1

Vi

+6Zw/ vJ] t
(4.26)

Bubbling in nonlinear ellipticproblems near criticality

since

k

E5,)~(U)-EE5,x(Uj)-fD

[

j=l

--zL -zL

£Vi6-

()6]

i=1

£Vi i=1

(2VoViVoVj + V/Vj + ViVi - 2)~e-4XViVj)

1<.l

t

(-4As2Vi - Vi" + Vi - 4Xe-4XVi)Vi

<,1

z
To estimate the quantities in (4.26), we consider the numbers 1

XI = ~(~l l +e~l), 1 ----2. . . . .

X1 = 0 ,

z

Xk+l ~ -~-0~,

k,

and decompose k Es,z(U) - Z

Es,z(Uj) = l<<.l~<~k,j>lfOnlx~
j=l

A straightforward computation yields B = o(e). On the other hand,

f)n{xI
vlSVJ l<~l<~k,j>l k I'XI+I = 4= ~=~ J x' W?WI+I -,}-o(8) _

=4~

_

f Xl+l wiSw~+l+o(e) '; XI

= 4rt f ×l+l--~l W S ( x ) W ( x -- (~l+1 Jx/-~l

-~1)) +

o(e)

k-1

eXW(x) 5 +o(e)

= 47~ Z e-I~/+t-~/I (12) 1/4 l=l

=

k-t/

F(.,+,I

a6 = l \ ~ - j

oo \1/2

/

+0(8).

283

284

M. del Pino and M. Musso

Taking into account the estimate given in Lemma 4.4 and the above estimates, we get (4.25) in the uniform sense. Similar arguments yield that the remainder is as well o(e) small after a differentiation with respect to the ~i'S. This concludes the proof. E3 REMARK 4.1. Single and multiple bubbling related with supercritical nonlinearity are also present in semilinear elliptic problem with Neumann boundary conditions. As we see below, what determines now the location of the bubble, or of the tower of bubbles in a domain with symmetries, is the presence of a nontrivial critical point situation for the mean curvature of OS2, that we denote by 7-/, with positive value. Let 1-2 be a bounded domain in ~U, N >~ 3, with smooth boundary Os and consider the boundary value problem -d Au + u = u p u>O ~Ou= 0

inS2, in S2, on OS2,

(4.27)

where p > 1 and d > 0. The works [58,69,70] have dealt with precise analysis of least energy solutions to this problem in the subcritical case, 1 < p < ~N+2 namely solutions which minimize the Rayleigh quotient

Q(u)-

d2 f ~ IVul 2 + fx~ lul 2 (fs? ]u] p+I)2/(p+I)

,

u E n 1(S2) \ {0},

(4.28)

for small d. From those works, it became known that for d sufficiently small, a minimizer ud of Q has a unique local maximum point xd which is located on the boundary. Besides, 7-/(xd) --+ maxxe0S2 7-/(x) and this solution decays exponentially which implies indeed the presence of a very sharp, bounded spike for the solution around xd. N+2 but Concentration phenomena of this type occurs as well in the critical case p - ~--~, with some important differences. Since compactness of the embedding of HI(I2) into L p+I (S2) is lost, existence of minimizers of Q(u) becomes nonobvious and in general not true for large d as established in [57]. It is the case however, as shown in [2,86], that such a minimizer does exist if d is sufficiently small. The profile and asymptotic behavior of this least energy solution has been analyzed in [3,68,79]. Again only one local maximum point xd located around a point of maximum for the mean curvature of 8S2 exists. However, unlike the subcritical case now its maximum value Md = ud(xd) --+ +cx~. Not only that, an important difference with the subcritical case is that now mean curvature is required to be positive at this critical point. In fact, nonnegativity of curvature is actually necessary for existence [4,47,79]. N+2 Consider now problem (4.27) when the power p is supercritical, namely p > ~--~-2" Recently in [ 19] it has been found that if N >~ 4, d is left fixed and one considers the exponent p as a parameter approaching the critical exponent from below, then single-bubbling solutions exist in certain cases. In particular, they find existence of single-bubble solutions with maximum points located on the boundary, near critical points of mean curvature with negative value.

285

Bubbling in nonlinear elliptic problems near criticality

Let the parameter d be fixed, with no loss of generality d = 1. In [36] we prove that, given a nontrivial critical point situation of the mean curvature of 0 Y2 with positive critical value, a solution exhibiting boundary bubbling around such a point of the following problem in ~ , in~2, on0~

- - A U + U -- U ( N + 2 ) / ( N - 2 ) + e

u>0 ~Ou= 0

(4.29)

exists, as e --+ 0, e > 0. Not only this: we are able to construct solutions with just one maximum point for which multiple bubbling is present. For instance if ~2 is a ball, there exists a solution whose shape is that of a tower, constituted by superposition of an arbitrary number of single bubbles of different blow-up orders. This phenomenon actually takes place just provided that ~ is symmetric with respect to the first (N - 1) variables, and 0 6 0Y2 is a point with positive mean curvature. Indeed, given k ~> 1, there exists for all sufficiently small e > 0 a solution ue of (4.29) of the form k (

u e (y )

:

OIN

i9~ 1

) (N-2)/2

1 1 +

)~26-2+(1-i)(4/(U-Z))]y[2

X ) (iN - 2) / 2 ~ - ( N - 2 ) / 2 -

i+1(1 -~- 0 (1))

for N >~ 4 and

ue(y)--o0 Z

1 +)~2e2-4illogel2ly[2

)~]/2e1/2-illoge[1/2(1 + 0 ( 1 ) )

i=1 q

for N -- 3, where o(1) ~ 0 uniformly in ~2 and/~i a r e explicit numbers. We would like to mention that existence of solutions to problem (4.29) which blow up at an interior point of the domain Y2 has been obtained in [80] in the case of dimension N -- 3 and in [81] for N ~> 4.

5. Liouville-type equations 5.1. Proof of Theorem 1.3 We present the construction of blowing-up families of solutions for problem (1.14) which lifts the nondegeneracy assumption of [9]. More precisely, we consider the role of nontrivial critical values of ~Pm in existence of solutions of (1.14), which is an example of nontrivial critical point situation (see [31]). Let D be an open set in ~m compactly contained in ;2 m with smooth boundary. We say that ~Om links in 7) at critical level C relative to B and Bo if B and B0 are closed subsets

286

M. del Pino and M. Musso

of 7) with B connected and B0 C B such that the following conditions hold: Let us set F to be the class of all maps 45 6 C (B, D) with the property that there exists a function qJ 6 C ([0, 1] x B, 7)) such that qJ (0, .) -- IdB,

~ (1, .) -- ,~,

qJ(t, ")18o = IdBo

for all t e [0, 1].

We assume

(5.1)

sup g)m(Y) < C =- inf s u p 9 9 m ( q ~ ( y ) ) , yEBo

cI)r

yEB

and for all y E 07) such that q9m (y) = C, there exists a vector ry tangent to OD at y such that (5.2)

V q g m ( y ) . r,y 7~ O.

Under these conditions a critical point y 6 7;) of q9m with q9m ( y ) = C exists, as a standard deformation argument involving the negative gradient flow of q)m shows. Condition (5.1) is a general way of describing a change of topology in the level sets {~0m ~< c} in D taking place at c = C, while (5.2) prevents intersection of the level set C with the boundary. It is easy to check that the above conditions hold if i n f q9m (x) < x~D

inf q)m (X) x~OD

or

sup q9m (X) > sup q9m (X), x~D

xCOD

namely the case of (possibly degenerate) local minimum or maximum points of ~0m. The level C may be taken in these cases respectively as that of the minimum and the maximum of q9m in 7). These hold also if 99m is C 1-close to a function with a nondegenerate critical point in D. We call C a nontrivial critical level of q9m in D.

THEOREM 5.1. Let m ~ 1 and assume that there is an open set 7) compactly contained in ~ m

where q9m has a nontrivial critical level C. Then, there exists a solution ue, with

lim ~2 fs eu~ -- 8m~.

~---+0

Moreover, there is an m-tuple (x~ . . . . . Xem) E 7), such that, as e --+ O,

. . . . . xm) --, 0,

q9m X . . . . .

Xm

~

C,

f o r which ue remains uniformly bounded on I-2 \ ujm=] B~(x[), and

sup u e ~

f o r any 6 > 0 .

+cx~

287

Bubbling in nonlinear elliptic problems near criticality

We will see that if X2 is not simply connected, such a set 79 actually exists for any m >~ 1, thus yielding the result of Theorem 1.3. For m = 1, a multiplicity result is also available: if X2 has d holes, then there exist at least d + 1 solutions ue, with lim 8 2 fs eu* -- 8ft.

8-+0

If X2 has d holes, namely d bounded components for its complement, then at least d + 1 solutions ue with lim 8 2 fs2 eu* = 8rt

8-+0

exist. We observe that (t91 (~) "-- H (~). Since H (~) approaches + o c as ~ approaches OX2, Ljusternik-Schnirelman theory yields that H has at least cat(X2) = d + 1 critical points with critical levels characterized through d + 1 min-max quantities. The same property is thus inherited for F (~) and the fact is thus established. We start providing an ansatz for solutions of problem (1.14). The "basic cells" for the construction of an approximate solution of problem (1.14) are the radially symmetric solutions of the problem Au + e" - 0 u(x) --+ - e c

in ]R2, as Ixl ~ oc,

(5.3)

which are given by the one-parameter family of functions 8/,t 2

col, (r) -- log (/Z 2 + r 2 ) 2 '

(5.4)

where # is any positive number. Let m be a positive integer and choose m distinct points in X2, say ~1 . . . . . ~m. Let/Z j, j = 1 . . . . . m, be positive numbers. We observe that the function

u j (x) = log

8/.z~

(Ix _ ~j[) +4log 1

(/Z~8 2 -+-IX -- ~jl2) 2 -- tOttj

8

7

satisfies in entire R 2 A u j -+- e2e uj -- 0. m

We would like to take E j = I Uj as a first approximation to a solution of the equation. We need to modify it in order to satisfy zero Dirichlet boundary conditions. Let Hj (x) be the solution of -AHj(x)

--O

Hj (x) -- - w # j (Ix-~jl) _ 4log

1

in X2, on 0 X2.

(5.5)

M. del Pino and M. Musso

288

We consider as initial approximation U - Eim=l (Uj -t- Hj), which by definition satisfies the boundary conditions. This approximation is less accurate n e a r ~j than u j alone unless Hj(~j) + Eim=l,ir + ui(~j)] '~ 0 as E ----> 0. We can achieve this by further adjusting the numbers/xj. As we will justify below, the good choice of these numbers is log 8# 2 =

H(~j, ~j) + E G(~l, ~j),

(5.6)

lCj where G and H are Green's function and its regular part as defined in the Introduction. Thus we consider the first approximation

U -i=1

[x--~i[) (ui + Hi) = ~1 ( (o)i 8

- log 84 + Hi ) ,

(5.7)

where O)i = O)#i and with the numbers//,j defined in (5.6). Let us analyze the asymptotic behavior of Hj as s --+ O. We observe that for x ~ 0 $2,

1 H j ( x ) = - 2 1 o g /-/,j2 s 2 + ] x - ~ j l 2 - 1 ~

8/z~

from where it follows that

Hi(x) =

H(x,~j)- log 8/z 2 + O(/z~s2),

(5.8)

uniformly in C2-sense for x on compact subsets of S2. Observe also that, away from each ~j, tOj = log 8# 2 + 4 log Ix - ~jl

and hence

Wj (X) + Hj (x) -- G(x, ~j) + O(82),

(5.9)

where the term O(.) is uniform in C2-sense on compact subsets of ~ \ {~j }. A useful observation is that u satisfies equation (1.14) if and only if

1 v(y) = u(sy) - 4 log E satisfies Av+e v-0 u >0 v=-41og? 1

inS2e, in S2~, onOl2~,

(5.10)

289

Bubbling in nonlinear elliptic problems near criticality

where s = s -1 $2. We also write ~[ 8-1~i and define the initial approximation in ex1 We want to measure how well V solves the panded variables as V (y) -- U (sy) - 4 log 7" above problem. Let us fix a small number 6 > 0 and observe that e v(y) = e4e U(x) with x -- ey, hence we see that =

k ( s y ) e v(y) - 0 ( 8

4)

if [ y - ~ } [ > - for all j 8

1 . . . . ,m.

(5.11)

Similarly, A V (y) = 8 2 A U (x) and (5.9) implies Ag(y)--0(8

4)

if ] y - ~1 > - for all j g

1 . . . . . m.

(5.12)

On the other hand, assume that for certain j , lY - ~.1 < ($. Then setting y -- ~ + z we get

eV(Y) =

(/z~ + Izl2) 2 x exp(Hj(~j

+ ~

+ sz)

log ~t, zE2

+ H~(~j + ez) .

lT~j

Now, by definition, log

I~ - ~j

14

+ H(~l, ~j) -- G(~I, ~j).

Taking into account this relation, the asymptotic expansion (5.8) and the definition of the numbers btl in (5.6) we get then that

8/s [1-1--0(8g)--[--0(82)] eV(y) = (bt~ + [y -- se} 12)2 '

ly-~jl,

6 < -" e

(5.1:3)

We also have in this region 8/s 2 A V (y) = A w~tj (lY - ~5 ]) + ~

= -

(/s jr_ [y __ ~.~ [2)2

+ O(s4).

(5.14)

In summary, combining (5.11)-(5.14) we have established the following fact: if we set R -- A V ( y ) + e v(y),

(5.15)

290

M. del Pino and M. Musso

then m

1

Ig(y)l ~
(5.16)

~j[3"

j=l

Let us set

W(y) - - e V(y). We have

m

W(y) - E

j=l

8/s

(lz~ 4- Jy

--~j

,12) 2 [1 4- Os(y)],

and Os has the property that, for some constant C independent of s,

m

[0s(Y)l ~
In terms of r problem (5.10) becomes L(r

"= A r 4- w e -- - [ R 4- N ( r

r

in S-2s, on 0 S-2s,

(5.17)

where N(r

= W[e r

1-r

(5.18)

A main step in solving problem (5.17) for small r under a suitable choice of the points ~j is that of a solvability theory for the linear operator L. In developing this theory we will take into account the invariance, under translations and dilations, of the problem Aw 4- e w - 0 in IR2. If we center the system of coordinates at, say ~., by setting z = y - ~j, then the operator L formally approaches the linear operator in IR2,

Lj(r

Ar

(~ff + Iz12)2 ~'

8~ namely, equation A v 4-e v = 0 linearized around the radial solution Vj (Z) -- log (#ff+lzl2)2. An important fact is the nondegeneracy of yj modulo the natural invariance of the equations under translations and dilations, ~ ~ vj (z - ~) and s ~-+ vj (sz) - 2 logs. Thus we

Bubbling in nonlinear elliptic problems near criticality

291

set 0

z~; (z) - ~ vj (z + ~) ogi ~=0

i-

1,2,

0

zoj(z) = -~s[Vj(SZ) + 2logs] s=l

It turns out that the only bounded solutions of Lj(d/)) = 0 in ]I~2 a r e precisely the linear combinations of the Zij, i -- 0, 1, 2, see [9] for a proof9 Let us denote also Zij (y) "-zij (y - ~j).

Additionally, let us consider a large but fixed number R0 > 0 and a nonnegative function X (P) with X (P) - 1 if p < R0 and X (P) - 0 if p > R0 + 1. We denote Xj(Y)--X(IY-~jl).

Given h of class C ~ (I2e), we consider the linear problem of finding a function q~ and scalars cij i = 1, 2, j - 1 . . . . . m, such that 2

m

in I2e,

(5.19)

~=0

on 0 S-2e,

(5.20)

fl2 xj Z i j ~ = 0

for all i -- 1, 2, j - 1 . . . . . m.

(5.21)

L(c/)) -- h + E

E

cij Xj Zij

i=1 j = l

8

Our main result for this problem states its bounded solvability, uniform in small e and points ~j uniformly separated from each other and from the boundary9 Thus we consider the norms -1

II~ll~-

sup I~(Y)I' yE~

I1~1]*-sup(~(l+lY-~.il) y~C2e j = l

-3+e2)

I~(Y)["

P R O P O S I T I O N 5 . 1 . Let ~ > 0 be fixed. There exist positive numbers eo and C, such that for any points ~j, j -- 1 . . . . . m, in 12, with

dist(~j, 0S2) ~> 5,

I ~ / - ~j[ >/3

for I ~ j,

there is a unique solution to problem (5.19)-(5.2 l ) f o r all e < eo. Moreover,

(5.22)

292

M. del Pino and M. Musso

Furthermore, the function ~' --+ 4) is C 1 and

1 )Ilh 11,. II0~s ~bl[~ ~< C log ~-

(5.24)

We observe that the orthogonality conditions in the problem above are only taken with respect to the elements of the approximate kernel due to translations. The proof of this result consists of some steps. The first step is to prove uniform a priori estimates for the problem (5.19)-(5.21) when ~b satisfies additionally orthogonality under dilations. Specifically, we consider the problem L(~b) = h

in I2e,

(5.25)

q$ = 0

on 0 S2~,

(5.26)

fs2 X j Z i j ~) = 0

for i

(5.27)

0, 1, 2, j = 1 . . . . , m,

and prove the following estimate. LEMMA 5.1. Let ~ > 0 be fixed. There exist positive numbers eo and C, such that f o r any points ~j, j -- 1 , . . . , m, in I2, which satisfy relations (5.22), and any solution to (5.25)-(5.27), one has I1~11~ ~ CIIhll,

(5.28)

f o r all e < eo.

PROOF. We will carry out the proof of the a priori estimate (5.28) by contradiction. We assume then the existence of sequences en --+ O, points ~j/7 E S2 which satisfy relations (5.22), functions hn with Ilhnll, ~ 0, 4~n with 114~nI1~ = 1, L(4)n) -- hn

in S2c,

(5.29)

4~n = 0

on 0 X2e,

(5.30)

fx_2xJ

Zijq~n--O

foralli = 0 , 1 , 2 , j -

1,...,m.

(5.31)

A key step in the proof is the fact that the operator L satisfies maximum principle in S2e outside large balls centered at the points ~j 9 Consider the function zo(r) = r2-1 radial l+r 2 , solution in 1R2 of 8

Az0 + (1 + r2) 2 z0 -- 0.

Bubbling in nonlinear elliptic problems near criticality

293

Define a comparison function in X2e, m

Z(y)--Ezo(alY-~}I),

y e $2c.

j=l

One can prove that if a is taken small and fixed, and R > 0 is chosen sufficiently large dem pending on this a, then we have that L(Z) < 0 in $?e "-- s \ ~ j = l B(~}, R). Since Z > 0 in this region, we then conclude that L satisfies maximum principle, namely if L(~r) ~< 0 in ~e and gr >~ 0 on 0~e then gr ~> 0 in ~e (see [35] for details). Let us fix such a number R > 0 which we may take larger whenever it is needed. Now, let us consider the "inner norm"

Ilell/-

sup 101. Uj~I B(~, R)

We make the following claim: There is a constant C > 0 such that if L(4)) = h in s

then (5.32)

IId~ll~ ~ cEIId~ll/+ Ilhll,].

We will establish this with the use of suitable barriers. Let M be a large number such that for all j , s C B(~}, 7-)" M Consider now the solution of the problem 2 - A ~ r j = ly - ~}[3 + 2e2

~rj(y) = 0

for

lY - ~ l -

_

t

R < [y

M

~J] < --8

M

R, lY - ~5]- e

A direct computation shows that

7r(r)

--

1

1

R

r

2 e (r-R)-

~1

LR

1

r

e2

-R

log(m/(eR))'

hence these functions have a uniform bound independent of e as long as 1 < R < ~ . On the other hand, let us consider the function Z (y) defined above, and let us set m

@(y) - 211011/Z(y) + Ilhll, ~

l/J'j(y).

j=l

Then, it is easily checked that, choosing R larger if necessary, L(0) <~ h, 0 ~> O on 0~E. Hence 0 ~< 0 on ~ . Similarly, 0 / > - 0 on S2~ and the claim follows. Let us now go back to the contradiction argument. The above claim shows that since II0~ I1~ - 1, then for some x > 0, II0~ Iii /> x. Let us set 0n(z) -- On(~] + z) where the

294

M. del Pino and M. Musso

index j is such that suPly_~},l
zx~+

(/z~ + Izl2) 2 q~ - 0.

This implies that ~ is a linear combination of the functions Zij, i -- 0, 1, 2. However, our assumed orthogonality conditions on ~bn pass to the limit and yield f X (Izl)zij~a - 0 and hence necessarily ~ -- 0, a contradiction from which the result of the lemma follows. [2] We want to establish next an a priori estimate for problem (5.25)-(5.27) with the orthogonality conditions f X j ~ Zoj = 0 dropped, namely the problem L(4~) = h

in S2e,

(5.33)

4~ = 0

on 0 S-2e,

(5.34)

fs2x j Zij~

--

0

for i - 1, 2, j - 1

m.

(5.35)

E

LEMMA 5.2. Let 6 > 0 be fixed. There exist positive numbers Eo and C, such that f o r any points ~j, j = 1 . . . . . m, in I2 which satisfy (5.22), and any solution ~ to problem (5.33)-(5.35), one has [[q~[l~ <~ C ( l o g ~ ) Ilhll,

(5.36)

f o r all e < eo. PROOF. Let R > R0 + 1 be a large and fixed number, and let Z0j be the solution of the

problem

AZOj +

(/z~ + ]Y - ~512)2 zoj -- 0, !

zoj(Y) = zoj(R)

for ]y - ~j[ = R,

"zoj(Y)-0

for l Y - ~}l = ~e"

A direct computation shows that this function is explicitly given by

r=ly-~}l.

Bubbling in nonlinearellipticproblemsnear criticality

295

Next we consider smooth cut-off functions 01 (r) and r/2(r) with the following properties: ~1 (r) - 1 for r < R, Ol(r) - 0 for r > R + 1, 1/71(r)l ~ 2; /72(r) -- 1 for r < 4~ , o2(r) = 0 ' l for r > ~ , [o2(r)l ~< Ce, Irl"2(r)l ~< Ce 2. Then we set

rllj(Y) = ~71(IY -- ~} I),

r/2j(y) -- r12([y - ~ l)

(5.37)

and define a test function

ZOj -- OljZoj +

(1 -

rllj)O2jZOj,

Zoj(Y)

--

zoj (lY - ~1)"

Intuitively, Z0j resembles the eigenfunction of the operator L associated to the invariance of L under dilations when L is considered in the whole IR2. Let 4) be a solution to (5.33)-(5.35). We will modify 4) so that the orthogonality conditions with respect to Zoj's are satisfied. We set m

j=l

where the numbers

dj are defined as

dJ fs ? xJlzojl2 + j ; xjZoj~ - 0 . E

E

Then m

L(~) -- h + ~--~djL(zoj),

(5.38)

j=l

and

fs2~ Xj Zoi~ = 0 for all i and all j. The previous lemma thus allows us to estimate

[

m

II~ll~ ~ c llhtl, + ~ldjlllC(~oj)ll,

] .

(5.39)

j=l

Testing equation (5.38) against z01 we find

(~, L(zo,)) - (h, zo,) + dl(L(zo,), ~:ol), where

(f, g) - fs2~ fg" This relation in combination with (5.39) gives us that m

.,(L(~o,),~o,) <. Cl,hll.[1 + IlL(~0,)II.] + c ~ Idjl [[L(~0j) I[~, j=l

(5.40)

296

M. del Pino and M. Musso

The following estimates hold true (see [35]): if we choose R sufficiently large, then C log(l/e)

IIL( oj ll,

(5.41)

and (L(z0t), ~'01) ~< -log(1/e---------~ 1 + O log(1/e-----~

"

(5.42)

Combining relations (5.42) with (5.40) and (5.41) we finally get that Idj.I ~< C

log ~-1)IIh II,

for all j = 1 . . . . , m. We thus conclude from estimate (5.39) that

114~II~ ~
The proof is complete. We are now ready for the proof of Proposition 5.1.

PROOF OF PROPOSITION 5.1. We begin by establishing the validity of the a priori estimate (5.23). The previous lemma yields

( 1t[

I1~11~~ c log-6

IIhll, +/~12 ~m 9

Icijl

1

(5.43)

,

j=l

hence it suffices to estimate the values of the c o n s t a n t s [cij [. Let us consider the cut-off function/72j introduced in (5.37). We test equation (5.19) against Zij 172j to find

(L(dp), rl2jZij)- {h, rl2jZij ) + cij fI2 xjlZijl2" 8

Now (t(~b), r/2j Zij)

- (~, t(172j Zij)).

We have

LO72jZij) -- Arl2jZij -k- 2Vrl2jVZij -+-eO((1

+ r) -3)

(5.44)

297

Bubbling in nonlinear elliptic problems near criticality

with r -

!

l Y - ~ j ] . Since A02j --O(82), V02j --0(8), and besides Zij = O ( r

1), VZij =

O(r-2), we find L(rl2jZij)

=0(83) -+-80((1 + r)-3).

Thus

1(4),L(o2jzij))l <~Csll?~lloo. Combining this estimate with (5.44) and (5.43) we obtain

Icijl ~ C

Iczkl

[ Ilhll, + ~ l o g -'~

1

8 l,k

1 which implies Icijl ~ CIIhll,. It follows finally from (5.43)that II~ll~ ~ C(log ~)llhll, and the a priori estimate has been thus proven. It only remains to prove the solvability assertion. To this purpose we consider the space

EHI(I2c): J~ xjZijdp-Ofori--l,2, j - 1 ..... m}, endowed with the usual inner product [4~, ~] = fs?~ V4~VO. Problem (5.19)-(5.21) expressed in weak form is equivalent to that of finding a ~b 6 H, such that for all 7t 6 H.

[~b, ~] = f ~ [-W~b + h]O dx

With the aid of Riesz's representation theorem, this equation gets rewritten in H in the operator form ~b = K(05) + h, for certain h ~ H, where K is a compact operator in H. Fredholm's alternative guarantees unique solvability of this problem for any h provided that the homogeneous equation 4~ - K (~b) has only the zero solution in H. This last equation is equivalent to (5.19)-(5.21) with h -- 0. Thus existence of a unique solution follows from the a priori estimate (5.23). We refer the reader to [35] for (5.24). D We recall that our goal is to solve problem (5.17). Rather than doing so directly, we shall solve first the intermediate problem 2

m

L(dp)---[R-q-N(dp)]nt-~~cijxjZij

in S2s,

(5.45)

i=1 j=l 4~-0

on 012s,

ff2xjZijdt)-O

foralli = 1,2, j -

(5.46) 1. . . . . m.

(5.47)

8

We assume that the conditions in Proposition 5.1 hold. We have the following result.

298

M. del Pino and M. Musso

LEMMA 5.3. Under the assumptions of Proposition 5.1 there exist positive numbers C and so, such that problem (5.45)-(5.47) has a unique solution d? which satisfies

I1r

~ Gel log sl.

Furthermore the map ~' ~-~ 4) into the space C(I2---e) is C 1 and the derivative D~,r defines a continuous function of ~f. Besides, there is a constant C > 0 such that

IID~,r

~ Csl log sl 2.

(5.48)

PROOF. In terms of the operator T defined in Proposition 5.1, problem (5.45)-(5.47) becomes r

T(-(N(r

+ R)) =~A(r

(5.49)

For a given number y > 0, let us consider the region Uy --= {r 6 C(~e)" [[r

~< ys[ logs[}.

From Proposition 5.1, we get

IIA(,)II~ ~ Cllog~l[[[g(,)]l, + tIell,]. Estimate (5.16) implies that IIRI[ ~< Cs. Also, the definition of N in (5.18) immediately yields

[IN(~)II, ~ cll,jl 2.

(5.50)

It is also immediate that N satisfies, for 4)1, ~b2 E .~c'?,, I I N ( ~ ) - N(qb2) 11, ~ Cyellogell]r

- ~b21l,,

where C is independent of y. Hence we get IIA(r

~< Cllogs]s[yZs]logsl 2 + 1],

IIA(Cj) -

A(r

~< Cys[ log e[21[r - r

It follows that, for all sufficiently small s, we get that A is a contraction mapping of 9t'• and therefore a unique fixed point of A exists in this region. For the dependence C 1 of r on the variable ~' and the estimate (5.48), we refer the reader to [35]. [5] After problem (5.45)-(5.47) has been solved, we find a solution to problem (5.17) and hence to the original problem if ~' is such that cij (~') = 0

for all i, j.

(5.51)

299

Bubbling in nonlinear elliptic problems near criticality

This problem is indeed variational: it is equivalent to finding critical points of a function of = e~'. To see that let us consider the energy functional Je associated to problem (1.14), namely

lfs ? IVu [2 dx - e2fs?eUdx.

Y~(u)- ~

(5.52)

F(~) - Je(U(~) + (6(~)),

(5.53)

We define

where U is the function defined in (5.7) and O~= 6 ( ~ ) - O~(x, ~) is the function defined x ~), with 4) the solution of problem (5.45)-(5.47) on I2 from the relation ~(x, ~) -4)(?-, given by Proposition 5.1. Critical points of F correspond to solutions of (5.51) for small e, as the following result states. LEMMA 5.4. Underthe assumptions of Proposition 5.1, the functional F(~) is of class C 1 Moreover, for all ~ > 0 sufficiently small, if D~ F (~ ) -- 0 then ~ satisfies system (5.51). PROOF. Define

1L

Ie(v) -- -~

[Vvl2dy -

L

(5.54)

eV dy.

oc

Let us differentiate the function F(~) with respect to ~. Since Je(U + cb) = Ie(V + d2), we can differentiate directly Ie (V + 4)) under the integral sign, so that

O~k,F(~ ) -- s - I D I s ( V + q~)[0$~. V -+- O~;,~b] 2

9

m

j--I

From the results of the previous section, this expression defines a continuous function of ~', and hence of ~. Let us assume that D~ F (~) = 0. Then 2

m

k-l,2,/-1 9

. . . . . m.

j=l

We recall that we proved IIDr ~< Cel log el 2, thus we directly check that as e --+ 0, we have Or + 0~4) - --[Zkl + o(1)] with o(1) small in terms of the L ~ norm as e ---> 0. We get that De F (r -- 0 implies the validity of a system of equations of the form m

2

#rci L 9

j=l

k-l,2,1-1,...,m,

300

M. del Pino and M. Musso

with o(1) small in the sense of the L c~ norm as e --~ 0. The above system is diagonal dominant and we thus get Cij = 0 for all i, j. This concludes the proof of the lemma. [] In order to solve for critical points of the function F, a key step is its expected closeness to the function J~ (U). LEMMA 5.5. The following expansion holds F(~) = J~(U) + 0~(~),

where

uniformly on points satisfying the constraints in Proposition 5.1. Furthermore, with the choice (5.6) for the parameters tzj, the following expansion holds Je(U) = - 1 6 m r t + 8mrt log 8 - 16mrt loge + 32rtqgm (~) + eOe(~),

(5.55)

where the function (19m is defined by (1.18). Here Oe is a smooth function of ~ -(~1 . . . . . ~m), bounded together with its derivatives, as e--+ 0 uniformly on points ~1 . . . . . ~m ~ 12 that satisfy dist(~i, 0 $2) > ~ and [~i -- ~j[ > 8. PROOF. Since/s (V) = J~ ( U ) , / s (V + O) = J~ (U + 0), it is enough to show that 0~ (~t) = 0e (e ~ ') satisfies

Taking into account DIe(V + 4~)[4~] = 0, a Taylor expansion gives

Ie(V

+

O)

-

Ie(V)

o 02Ie(V

--

fol(f

+

t~b)[~b]2(1

[U(~b) + R]0 +

t)dt

)

k(ey)eV[1-etO]~ 2 (1 - t) dt.

(5.56)

Since I1~11~~ GEl log el, we get

Ie(V + qb) - Ia(V) = O~ - O(e21 log el3). Let us differentiate with respect to ~t. We use the representation (5.56) and differentiate

Bubbling in nonlinear elliptic problems near criticality

301

directly under the integral sign, thus obtaining, for each k - 1, 2, l - 1, . . . , m,

O~s --

+ cb)-

fo'

Using the fact that we get

Is(V)]

0,1,, [ ( N ( r

)

a,s

R),] +

8

- et~]4b2 ] (1 - t ) d t .

8

I10~,~11,~ Gel log sl 2 and the computations in the proof of Lemma 5.3

O$i,.,[Is(V + 4 ) ) - Is(V)] - O$s Os -

o(sel logsl4).

The continuity in ~ of all these expressions is inherited from that of q$ and its derivatives in ~ in the L e~ norm. To obtain (5.55), we just mention that the following asymptotic expansions hold true

IVUI 2 dx - - 8 m r t +

16rt log j=l

j=l

8/Zj

i#j

and 82

fs2 k(x)eU dx = 8mrt + sOs(~). [3

For the details, see [35].

PROOF OF THEOREM 5.1. Let us consider the set 79 as in the statement of the theorem, C the associated critical value and ~ 6 79. According to Lemma 5.4, we have a solution of problem (1.14) if we adjust ~ so that it is a critical point of F(~) defined by (5.53). This is equivalent to finding a critical point of F(~) -- F(~) + 16m~t log e. On the other hand, from Lemma 5.5, we have that for ~ E D, such that its components satisfy [~i - ~j[/> 5,

o~F (~ ) + fl --(flm(~)

+

8(~s (~ ),

where Os and V~ Os are uniformly bounded in the considered region as s -+ 0, and ot 7~ 0 and 15 are universal constants. Let us observe that if M > C, then assumptions (5.1), (5.2) still hold for the function min{M, qgm(~)} as well as for min{M, q)m(~) + eOs(~)}. It follows that the function

M. del Pino and M. Musso

302

min{M, otF(~) + fl} satisfies for all s small assumptions (5.1), (5.2) in 79 and therefore has a critical value Cc < M which is close to C in this region. If ~ 6 79 is a critical point at this level for ot F (~) + fl, then since N

N

oeF(~s) + fl ~< Ce < M, we have that there exists a 3 > 0 such that I~e,j -~e,i[ > 3, dist(~e,j, 0~2) > 0. This implies Cl-closeness of otF(~) + / 3 and qgm(~) at this level, hence Vqgm(~e) -+ O. The function u~ -- U ( ~ ) + 4)(~) is therefore a solution as predicted by the theorem. [-q PROOF OF THEOREM 1.3. According to the result of Theorem 5.1, it is sufficient to establish that given m >~ 1, (,/9m has a nontrivial critical value in some open set 79, compactly contained in S2m. Our choice of 79 is just given by 79 = {y 6 x2m I dist(y, 0,('2rn)

>

3},

where 3 is a small positive number yet to be chosen. We observe that in this set function Y~-j=I H(yj, yj) is bounded and ~-.ig=j G(yi, yj) is bounded below. Consequently function qgm(y) is also bounded below in 79. Let ~1 be a bounded nonempty component of ]~2 \ ~, and consider a closed, smooth Jordan curve y contained in $2 which encloses a'21. We let S to be the image of y, B0 = 0 and B = S x ... • S = Sm. Then define m

C-- inf sup qgm(q:' (Z)) , ~F

(5.57)

zEB

where 45 E /-' if and only if q ) ( z ) - ~ ( 1 , z) with !P" [0, 1] x B -+ 7) continuous and q-' (0, z) = z. We first need the following lemma. LEMMA 5.6. There exists K > 0, independent of the small number 3 used to define 7) such

that C >~- K. PROOF. We need to prove the existence of K > 0 independent of small 3 such that if q~ 6 F , then there exists a ~ 6 B with

qgm(~(Z,)) ~--K.

(5.58)

Let us assume that 0 6 S21 and write

(i0 (Z) -- (~1 (Z) ..... (JDm(Z)). Identifying the components of the above m-tuple with complex numbers, we shall establish the existence of ~ 6 B such that cloj(~)

Ir (~)1

__ e2jrti/m

for all j = 1 . . . . . m.

(5.59)

Bubbling in nonlinear elliptic problems near criticality

303

Clearly in such a situation, there is a number/z > 0 depending only on m and $2 such that

1r (Z) -- r (Z)[ ~> /s This, and the definition of qgm clearly yields the validity of estimate (5.58) for a number K only dependent of S-2. To prove (5.59), we consider an orientation-preserving homeomorphism h" S 1 --+ S and the map f " T m ~ T m defined as f ( ( ) - - ( f l ( ( ) . . . . . f m ( ( ) ) with Tm~s

1 •

• S1 m

and

qOj (h((1) . . . . . f j ( ~ l . . . . , ~m) =

We define a homotopy F ' [ 0 , 1] • T m ~

Fj(t,()=

h((m))

I ~ j ( h ( ( 1 ) . . . . . h((m))l

~j(t,h((1)

T m by

..... h((m))

Iq~j(t, h((1) . . . . . h ((m)) I

Notice that F (1, () -- f (() and

F(0, () =

(hr162 Ih(~~)l

h(~m)) Ih((m)l

which is a homeomorphism of T m . The existence of ~ such that relation (5.59) holds follows from establishing that f is onto, which we show next. The toms T m can be identified with the closed manifold embedded in •m+l parameterized as

if'(01 . . . . .

Om) E

[0, 2~) m ~-~ (pl ei0' , 0 m - l ) + (01, p2 ei02, 0m-2) + ' ' " + (0m-l,

tom eiOm),

where 0 < Pm < "'" < /91 and we have denoted Ok -- (0, . . . , 0), e iOj - - (cos Oj , sin Oj ) . We

k consider as well the solid toms T m parameterized as (01 . . . . . Om, p ) ~ [0, 2rt) m • [0, Pro] ~

(Pl ei0' , 0 m - l ) -~- (01, p2 ei02, 0m-2) +

Obviously OAm T -- T m in ~Km+l.

...

-]-- (0m- 1 , p e iOm ).

304

M. del Pino and M. Musso

With slight abuse of notation, we consider the map f ' T m --+ T m, induced from the original f under the above identification, namely

f ( ( ) = (Plfl (~'), Om-1) -F (01, P2f2(~'), Ore-2) -Jr-"'-Jr- (Om-1, Pmfm(()). The function f then can be extended continuously to the whole solid torus as 3~" Am T --+ Rm+l defined simply as

f ( s r, P) = (Plfl (~'),

Om-1) -Jr (01, P2f2(~'), Om-2) -1-""" "1- (Om-1, Pfm(~)).

The function f is homotopic to a homeornorphism of ~'m, along a deformation which applies OT m into itself. Thus if P ~ int(T m) then deg(f, T m, P) 7/= 0 and hence there exists Q ~ r m such that f ( Q ) = P. Thus if we fix angles (0~ . . . . . 0") E [0, 2n:) m and p* ~ (0, Pm) then there exist ~'** ~ T m and p** ~ (0, Pro) such that A

A

(,01fl (~'**), Om-1) -Jr-(01 ,02f2(~'**), Om-2) -+-

+ (Om-1

fm((**))

-- (ple i0:, 0m-l) -k- (01, p2 ei0;, 0m-2) -+-""" 4- (0m-l, p*ei0m*). A direct computation shows then that f j (~**) -ei~ < for all j and also p* - p**. It then follows that f is onto. This concludes the proof. E] The second step we have to carry out to make Theorem 5.1 applicable is to establish the validity of assumption (5.2). To this end we need to establish a couple of preliminary facts on the half-plane

~-~-- {(xl,x2)" X1 ~ 0}. LEMMA 5.7. Consider the function o f k distinct points on 7-I qJk(xl , . . . , xk) = --4 Z

log IXi -- Xj [.

ir Let I+ denote the set o f indices i f o r which x) > 0 and Io that f o r which xli - O. Then, either

Vxi tlJk(X 1. . . . . Xk) r 0

f o r some i ~ I+

or ~~k(Xl

OXi2

. . . . . Xk) r 0

f o r some i ~ I0.

Bubbling in nonlinear elliptic problems near criticality

305

PROOF. We have that 0 - - tPk ()~Xl . . . . . )~Xk) 0)~

)~=1

= Z Vxi tlIk(Xl . . . . . Xk)Xi -4- Z aXi2tIJk(Xl . . . . . Xk)Xi2. ieI+ i61o

On the other hand, 0 ~ @k ()~Xl . . . . . )~Xk)

0 [k(k - 1)log)~] ~__m4m

r X=I

)~=1

D

and the result follows.

A second result we need concerns the analogue of the function q)k, for the half-plane 7J. Let x = (x I , x2), y = (yl, y2). Then regular part of Green's function in 7-[ is now given by H (x, y) = - 4 log

~ _ (yl, _y2). Ix - YI'

Then 1

G(x, y) = 4 log

4 log

Ix - yl

1

Ix-Yl

Hence the associated function qSk is given by k q)k(Xl

.....

Xk) -- 4 ~ i=1

1 Ixi --xj[ log ~~i---------~ [ x__ i -Jr-4 ~ log [xi - x j l ir

With identical proof as the previous lemma we now get the following one. LEMMA 5.8. For any k distinct points xi E int(7-/) we have vqSk(Xl . . . . . xk) r

We will recall here some straightforward to verify facts about the regular part of the Green function H ( x , y) -- G(x, y) - 4 log ix_--lyl. Let y 6 s be a point close to 012 and let be its uniquely determined reflection with respect to 0 12. Set O(x, y) = H ( x , y) + 4 log

Ix - Yl

306

M. del Pino and M. Musso m

Then it can be shown that 7t (x, y) is bounded in s x I-2 and

]vxg~(x, y)[ § [Vyg~(x, Y)I ~
(5.60)

Using (5.60) one can derive the following estimates

[ V x H ( x , y)[ + ] V y H ( x , y)[ ~< C1 min

{ Ix -, y[' dist(yl, 012) } + C2.

(5.61)

Now we are ready to prove the validity of assumption (5.2) which in this case reads as follows: LEMMA 5.9. Given K > 0, there exists a 6 > 0 such that if (~1 . . . . . ~m) ~ 079 and [qgm(~1 . . . . . ~m)[ <~ K , then there is a vector r, tangent to 079 such that V~m (~1 . . . . , ~m)" Z" r 0.

PROOF. Let us assume the opposite, namely the existence of a sequence 8 --+ 0 and of points ~ -- ~a for which ~ ~ 079 and such that V~i qgm(~1 . . . . . ~m) -- 0

if ~i E a"~a

(5.62)

and (5.63)

if ~i E 0~(-2~,

V~iqgm(~l, . - . , ~m)" "gi -- 0

for any vector ri tangent to OS2a at ~i, where S2a - {x E I2- dist(x, O1-2) > ~ }. From the assumption of the lemma it follows that there is a point ~l ~ Ol2~, such that H(~l) --+ --oo as 6 --+ 0. Since the value of q9m remains uniformly bounded, necessar1 ily we must have that at least two points ~i and ~j that are becoming close. Let 6n -- n' ~n _ ( ~ . . . . . ~m) ~ S2an be a sequence of points such that (5.62), (5.63) hold, and Pn = inf I~i'"., -- ""'~:I --+ 0 iT~j

as n --+ ec.

Without loss of generality we can assume that Pn - I ~

- ~ I. We define

/7

xj = ~ . Pn

(5.64)

Clearly there exists a k, 2 ~< k ~< m, such that lim Ix j[ < e c , n -----~o43

j = 1 .....

k,

and

lim n---+ oo

For j ~< k we set .~j -- lim x j/7. n---~ o~

Ix l-

J~

k

Bubbling in nonlinear elliptic problems near criticality

307

We consider two cases 9 (1) either dist(~f, Of2~. ) -----~ OO,

Pn

(2) or there exists co < ec such that for almost all n we have

< CO.

Pn C A S E 1. It is easy to see that in this case we actually have

dist(~j, a f2~. ) --+ oc,

j - - 1 . . . . . k.

Pn

Furthermore points ~ . . . . . ~ are all interior to S2~,, hence (5.62) is satisfied for all partial derivatives 0@, j ~< k. Define q)m(Xl . . . . .

X m ) - - q)m (~1 + p n X l . . . . .

~1 -Jr- P n X m ) .

We have for all 1 - 1, 2, j -- 1 . . . . . k,

Then at 2 - - (21 . . . . . xk, 0 . . . . . O) we have

OXljq)m (~C) - -

O.

On the other hand, using (5.61) and letting p, --+ 0, we get lim

pnO@qgm(~ +xpn) - -4 Z

n-+c~

Ox,;log

i=/:j,i <~k

1

=0.

[~Cj -- Xi [

Since this last equality is true for any j ~< k, 1 -- 1, 2 we arrive at a contradiction with L e m m a 5.7 which proves impossibility of the Case 1. It remains to consider the second case. C A S E 2. In this case there exists a constant C such that

dist(~, 0s

) ~C,

j-1

. . . . ,k.

Pn

If there points ~] are all interior to I2~, then after scaling with Pn we argue as in Case 1 above to reach a contradiction with L e m m a 5.8.

M. del Pino and M. Musso

308

Therefore, if Case 2 is to hold, we assume that for certain j = j* we have dist(~jn , 0 s

- O.

Assume first that there exists a constant C such that (~n ~ CPn. Consider the following sum (summation here is taken with respect to all i 7~ j ) s,

=

i:/:j

The leading part of Sn, as n --+ cx~, comes just from the points that become close as n --+ 0. We can isolate groups of those points according to the asymptotic form of their mutual distances. For example, we can define

iT~j,i,j>k

and consider those points whose mutual distances are O(p 1), and so on. For each group of those points (also those with indices higher than k) the argument given above in the Case 1 applies. This means that not only those points become close to one another but also that their distance to the boundary 0s is comparable with their mutual distance. Applying the asymptotic formula for the Green's function we see that Sn -- O(1)

as n ~ oc.

(5.65)

On the other hand, we have

Z H (~j,

~'.~) ~< H (,,ej,,

~jn) + C ~< - 4

log

+C.

J

Since [~jn _ ~jn[ ~< 26n (because ~jn, E O.Q~n), we have that

a s / ' / - - + 0Q,

J which together with (5.65) contradicts the fact that qgm(~n) is bounded uniformly in n. Finally assume that Pn -- O(•n). In this case after scaling with Pn around ~jn and arguing similarly as in the Case 1 we get a contradiction with Lemma 5.7 since those points ~j that are on Os after passing to the limit, give rise to points that lie on the same straight line. Thus Case 2 cannot hold. fl In summary we reached now a contradiction with the assumptions of the lemma. The proof of Theorem 1.3 is complete. V1

Bubbling in nonlinear elliptic problems near criticality

309

REMARK 5.1. Let us mention that in [27,28] we study some elliptic problems in a twodimensional domain with nonlinear Neumann boundary condition where the nonlinearity is exponential on the boundary. Indeed, in [28] the problem Au - - 0

in S-2, (5.66)

OU

Ov

= ee u

on 0S2

was analyzed. In (5.66), 12 is a bounded domain in ~2 with smooth boundary and e a positive small parameter. We prove that given any domain S2 and any k >~ 1, for e sufficiently small, a couple of positive solutions peaking at k points on the boundary of S-2 in such a way that )~eu ~ 4rt )--~=1 6~j was built up, using as basic cells (after suitable zooming-up) explicit solutions of Av - 0 OU

= ev

Ov

in R 2, 9

on 0 k _-1-,

where R 2 denotes the upper half-plane {(Xl, X2)" X2 > 0} and v the unit exterior normal to OR~_, given by

2# tOt/z (Xl, X2) - - log (Xl - t) 2 + (x2 -Jr-/z) 2 '

(5.67)

where t 6 R and/z > 0 are parameters. These functions are the basic cells to build solutions to a related problem with nonlinearity of exponential type on the boundary, namely Au - - 0

in S2, (5.68)

OU

Ov

= 2)~ sinh u

on 0S2.

In [27] we prove that in any domain S2 and for any k ~> 1 there are at least two distinct families of solutions to (5.68) which exhibit exactly the qualitative behavior of the explicit solution (5.67) at 2k points of the boundary and with alternate signs. See also [89] for a related problem.

5.2. A related 2-d problem involving nonlinearity with large exponent In what is left of this section we deal with the analysis of solutions to another twodimensional nonlinear elliptic problem, namely the following boundary value problem Au+uP--O u>0 u--0

in S-2, in S'2, on 0S2,

(5.69)

310

M. del Pino and M. Musso

where S2 is a smooth bounded domain in ]t~2 and p is a large exponent. As we will see, problem (5.69) shares similar patterns with the one discussed above in this section, problem (1.14). First observe that since H I (I2) is compactly embedded in L p+I (S2) for any p > 0, standard variational methods show that Sp, given by

Sp =

inf Ip(u), ,~H~ (;2)\{0}

where Ip denotes the Rayleigh quotient

fs2 IVul 2 Ip(u)-

(fs2 [u[P+l)2/(p+l)'

u E HI(S-2) \

{0},

is achieved by a positive function Up which solves problem (5.69). This is known as least energy solution. In [76,77] the authors show that the least energy solution has Lee-norm bounded and bounded away from zero uniformly in p, for p large. Furthermore, up to subsequence, the renormalized energy density plVupl 2 concentrates as a Dirac delta around a critical point of Robin's function H(x, x) introduced in (1.19). In [1 ] and [39] the authors give a further description of the asymptotic behavior of Up, as p --+ ec. Indeed they prove that IlUpIlee ~ ~/e as p --+ ec and that the limit profile of these solutions, properly translated and normalized, is given by the radially symmetric solution (5.4) to problem (5.3). Hence problem (5.3) is what in literature is known as "limit problem" not only for problem (1.14). Indeed (5.3) is the limit problem associate to (5.69) too. In the same spirit of the problems discussed above, we can build solutions for problem (5.69) that, up to a suitable normalization, look like a sum of concentrated solutions for the limit profile problem (5.3) centered at several points ~l . . . . . ~m as p --+ ec. At this point, another analogy with problem (1.14) appears: the function responsible to locate the concentration points ~1 . . . . . ~m is the same function q9m defined by (1.18), which is responsible for the location of the concentration point for the Liouville-type problem (1.14). Our main result, contained in [41], guarantees that, as soon as I-2 has a hole, with no restriction on the size of the hole, then problem (5.69) admits a solution with an arbitrary number of point of concentration. This is the result contained in the following theorem. THEOREM 5.2. Assume that S2 is not simply connected. Then given any m ~ 1 there exists Pm > 0 such that for any p ~ Pm problem (5.69) has a solution Up with

l i m p fs? upp+I -- 8rtm.

p--+oc

As for the Liouville equation (1.14), the previous result can be obtained as a consequence of the following more general theorem.

Bubbling in nonlinear elliptic problems near criticality

311

THEOREM 5.3. Let m >~ 1 and assume that there is an open set 79 compactly contained in if2 m where q9m has a nontrivial critical level C. Then, there exists Pm > 0 such that f o r any p >1 Pm problem (5.69) has a solution Up which concentrates at m different points o f I2, i.e., as p goes to + ~ , m

p u p+l ~ 8rte ~ 6~j i=1

weakly in the sense o f measure in $2

(5.70)

f o r some ~ E 79 such that q9m (~1 . . . . . ~m) -- C and Vq9 m (~1 . . . . . ~m) -- O. More precisely, there is an m-tuple ~ p - (~P . . . . . ~P) ~ 79 converging (up to subsequence) to ~ such that, f o r any 6 > O, as p goes to +zx~, up -+ 0

uniformly in S2 \

ujm__l Bs(~/p)

(5.71)

and

sup Up(X) --+ x/-e. xcS~(~p)

(5.72)

The proof of how Theorem 5.3 implies the result contained in Theorem 5.2 is identical as the one to prove Theorem 4.4. As already mentioned, the case of (possibly degenerate) local maximum or minimum for ~m is included. This simple fact allows us to obtain an existence result for solutions to problem (5.69) also when S2 is simply connected. Indeed, we can construct simplyconnected domains of dumbbell type where a large number of concentrating solutions can be found. Let h be an integer. By h-dumbbell domain with thin handles we mean the following: let $2o -- 121 U ... U S2h, with 121 . . . . ,12h smooth bounded domains in ~ 2 such that S-2i O ,.(-2j -- ~ if i r j. A s s u m e that ~('2i C { (Xl, X2) E ]I~2" ai ~ Xl ~ hi },

if2 i 0 {x2 -- 0} ~;&~,

for s o m e bi < ai+l and i -- 1 . . . . . h. Let

Ca --

{ (Xl, x2) E ]R2" Ix2[ ~ e, Xl E (al,

bh) }

for some e > 0.

We say that 12~ is an h-dumbbell with thin handles if S-2~ is a smooth simply connected domain such that S-20 C l-2e C $20 U Ce, for some e > 0. The following result holds true. THEOREM 5.4. There exist eh > 0 and Ph > 0 such that f o r any e e (0, eh) and p ~ Ph problem (5.69) in 12~ has at least 2 h - 1 families o f solutions which concentrate at different points o f $2~, according to (5.70)-(5.72), as p goes to + ~ . More precisely, f o r any integer 1 <<,m <<,h there exist (hm)families o f solutions o f (5.69) which concentrate at m different points o f l-2c.

312

M. del Pino and M. Musso

The detailed proof of how Theorem 5.3 implies the result contained in Theorem 5.4 can be found in [40]. The proof of all these existence results relies on a Lyapunov-Schmidt procedure, based on a proper choice of the ansatz for the solution we are looking for. Usually, in other related problems of asymptotic analysis, and in particular in all the problems discussed in this chapter, the ansatz for the solution is built as the sum of a main term, which is a solution (properly modified or projected) of the associated limit problem, and a lower-order term, which can be determined by a fixed point argument. In this specific problem, this approach is not enough. Indeed, in order to perform the fix point argument to find the lower-order term in the ansatz (the equivalent version of Lemma 5.3 for problem (5.69)), we need to improve substantially the main term in the ansatz, adding two other terms in the expansion of the solution in order to improve the order of the error from p - 2 to p-4. This fact is basically due to the fact that, when one write the equation in (5.69) as L~b = - R - N(~p), with L a linear operator, R the error for the first approximation to be an actual solution to the problem and N(~b) a quadratic term in q~, one only gets Cpllq~ll 2oo

instead of the expected CIl ll 2

as in (5.50). By performing a finite-dimensional reduction, we find an actual solution to our problem adjusting points ~ inside S'2 to be critical points of a certain function F (~) (the equivalent to (5.53) for problem (5.69)). It is quite standard to show that this function F(~) is a perturbation of qgm(~) in a C~ On the other hand, it is not at all trivial to show the C 1 closeness between F and qgm. This difficulty is related to the difference between the exponential decay of the scaling parameters ~ ~ e - p / 4 and the polynomial decay 1 of the error term IIA U~ + Uff ]l, of our approximating function U~. We are able to overcome this difficulty using a Pohozaev-type identity. For the detailed proof of all these results on problem (5.69) we refer the reader to [41 ] (see also [42] for some results concerning changing sign solutions).

Acknowledgement This work has been partly supported by grants Fondecyt 1030840, Fondecyt 1040936 and FONDAP, Chile.

References [1] Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc. 132 (4) (2004), 1013-1019.

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