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Some concentration phenomena in degenerate semilinear elliptic problems
NonlinearAnalysis, Theory,Methods& Applications,Vol. 24, No. 7, pp. 1011-1025, 1995 Copyright © 1995ElsevierScienceLtd Printed in Great Britain. All rights reserved 0362-546X/95 $9.50+ .00
Pergamon 0362-546X(94)00127-8
SOME CONCENTRATION DEGENERATE SEMILINEAR
PHENOMENA IN ELLIPTIC PROBLEMS
DONATO PASSASEO Dipartimento di Matematica, Universitfi di Pisa, Via Buonarroti 2, 56127-Pisa, Italy
(Received 16 July 1993; received f o r publication 31 May 1994) Key words and phrases: Nonlinear Dirichlet problems, degenerate equations, concentration p h e n o m e n a and nonexistence results, multiplicity of positive solutions.
0. I N T R O D U C T I O N
Let ~ be a smooth bounded domain of R ~, )t ~ L®(~) be a function almost everywhere positive in D and g: ~ ~ ~ be a continuous function such that g(0) = 0 and as t . ~ +_oo,
tg(t) - [ t f
w i t h p > 2 a n d p < 2 n / ( n - 2) if n > 3. In this paper we study the existence and the multiplicity of nontrivial solutions u (i.e. u ~ 0 in ~) for the problem
(*)
I div(;tDu) + g(u) = 0
in f~
(. u = 0
on 0t~.
Well-known results guarantee existence and multiplicity of (positive or nodal) solutions for problem (,) in the case that infA > 0 (see, for example, [1, 2]). u
On the contrary, several problems arise when the equation is degenerate, that is inf 2 = 0: in this case the classical topological methods of the calculus of variations cannot be applied directly and the existence and multiplicity results, which hold when i n f 2 > 0, cannot be extended to the degenerate case. Many papers have been devoted to the study of several questions concerning degenerate elliptic problems (see, for example, [3-5] and the references therein). In this paper we study some concentration phenomena for the solutions of problem (.), which recall some situations analogous to those occurring in elliptic problems with critical or supercritical Sobolev exponents (see [6-16]). We show that, as in those problems, existence, nonexistence and multiplicity of nontrivial solutions for problem (.) are closely related to these phenomena. In our case, such concentration phenomena are due not to the Sobolev exponent 2 n / ( n - 2), but to the degenerate character of the differential equation, as shown by the following example: let ~,(x) = Ix[" with c~ > 0 and suppose that 0 e ~ and g(t) tltlP-2; as is well known, for a function g of this type, the nontrivial solutions of problem (,) correspond to the critical points of the functional =
f ( u ) = I ~.(x)[Du[ 2 dx J fi lOll
1012
D. PASSASEO
constrained on the manifold
I If o~ < (2n + 2p - np)/p, then there exists the minimum o f f on ~ and so problem (.) has a (positive) solution (see theorem 2.2); on the contrary, if ce > (2n + 2p - np)/p, then this minimum does not exist because i n f f = 0, as one can easily verify (it suffices to put
vp ¢,(x) = (o(x/e) with ¢~ e C~(B(O, 1)) and to remark that lim f(~,/]lgo, l],) = 0). e~0 +
Moreover, the results proved in this paper (see theorem 1.3) show that, if f~ is starshaped with respect to 0 and ct _ (2n + 2p - np)/p, not only does the minimum o f f on the constraint Vo not exist, but there is not even a constrained critical point. On the contrary, if 2 is infinitesimal of order small enough (in the sense that 1/2 e if(g)) for a suitable q > 1), then one can prove a compactness property (see proposition 2. l) which enables one to obtain general existence and multiplicity results as in the case inf A > 0. n In the problems with critical and supercritical exponent considered in [6, 7, 9, 12-22], as well as in the ones studied in [23-26], some concentration phenomena like these enable one to relate the number of positive solutions to the shape of the domain f~; in an analogous way, it is possible to relate the multiplicity of positive solutions of problem (,) to the properties of the function 3., as shown by the results stated in Section 3 (see theorem 3.2 and remark 3.3). The paper is organized as follows. In Section 1, we prove an identity (like the Pohozaev identity [6]) which gives some conditions on functions ~. and g, sufficient to exclude the existence of nontrivial solutions for problem (.) in starshaped domains; in Section 2, on the contrary, we give sufficient conditions for the existence and the multiplicity of nontrivial solutions for problem (.); in Section 3, we study the effect of the properties of function 2 on the number o f positive solutions of problem (,): we consider a family of functions (~-,),>o, such that inf 2~ > 0
ve > 0
fl
and
lim inf 2~ = O. ,~0 +
fl
For these uniformly elliptic problems P~, which approximate degenerate elliptic problems, we find some solutions which, as e -~ 0 +, tend to "retire itself" in the subset of ~ where ~-e ~ 0; hence, if the subset where ~ ~ 0 is constituted by k connected components, we prove the existence o f at least k + 1 positive solutions for e > 0 small enough, and we describe their qualitative properties and the asymptotic behaviour as e ~ 0 + (see theorem 3.2 and remark 3.3). Let us note that the nonexistence results of Section 1 and the existence results given in Section 2 are complementary and describe completely the situation: indeed, the example considered above (2(x) = Ixl% g(t) = t It [p-2, n starshaped with respect to 0) satisfies the assumptions of theorem 1.3 or those of theorem 2.2; hence the assumptions considered in these two theorems are optimal. On the contrary, theorem 3.2 is stated in a simplified form; some of its possible generalizations have been pointed out in remark 3.3. 1. N O N E X I S T E N C E
RESULTS
In this section we prove that the solutions of problem (.) verify a general identity which, in a particular case, recalls the Pohozaev identity (see [6]), and is very important in studying
Concentration phenomena
1013
elliptic problems with critical and supercritical Sobolev exponents (see lemma 1.1 and corollary 1.2). This identity gives a condition on functions ). and g, which is sufficient to exclude the existence of nontrivial solutions in starshaped domains D (see theorem 1.3). LEMMA 1.1. Let D be a smooth bounded domain of ~ ' , ;t be a smooth real function almost everywhere positive in D, g: R --, ~ be a continuous function and u be a solution of the problem
l div(;t(x)Du(x)) + g(u(x)) =
0
u = 0
in D on OD.
Then, for every vector field v e CI(~, ~'), it results in
1 I ~ Di2vilDul2dx,
2 ~i=1
where v denotes the outward normal to the boundary of t) and
Proof. For every w e
G(t) -- I[ g(r) dr.
cl(~r~, [~) one has
f div(2Du)wdx+In g(u)w dx = O, that is fan'~w(Du'v)dtr-f9
2DuDwdx + 19 g(u)wdx : O.
In particular, if for a fixed v e C1(~, ~ ' ) we set
Du = (Du. v)v on the boundary of £))
w = (Du'v),
then we have (since
lan A'Dul2(v " v) da = fg "~j~IDiUDi( i~l DiUvi) dx - t'g g(u)( i~=l DiUvi) dx, that is
f 09 XlDul2(v"v) da= I 2fl i,j~=1DjuDijuvidx+ f fl i,j=l DjuDiuDjvi dx - f9 g(u) i=l ~ Diuvidx and so l
o. ;tlDul2(v"
~Di(lDul2)vidx+ f 9 i,j=l ADivjDiuDju dx
v) d a = ~I I 9 A i = ,
-- ig i=~ l Di[G(u)I vidx'
(1)
1014
D. PASSASEO
which implies
12 on ;qDul2tv • v) d a = - 2 ,1j f l i = l~ Di()tVi)lDu[ 2 dx +
2DiujDiuDju dx
+ ( G(u) div v d r , J that is
1 2 oa
~lOuff(v'v) d a = - ~ 1 .i=~ ~, Dido, IDul 2 dr - ~
fl i , j = 1
[~i=l
~Div, IDul2 dr
fl
= fa [ G(u) - 2 ~D-~] div(v)dx + laA
i,j=l
DivjDiuDju dx
1 I IDul2 ~ Di;tvidx.
2 a
i=1
In particular, if in the previous lemma we set v(x) = x, we obtain for the solutions of problem (,) a relation analogous to the Pohozaev identity (see [6]). COROLLARY1.2. Let ~, ~., g verify the same assumptions of lemma 1.1. Then, for every solution of problem I div(2Du) +
g(u) = 0
in g)
u = 0
on O~
we have (with the same notations of lemma 1.1)
1
2 on
;~lDul2(x • v)da
-- n
S
n
G(u)dx +
a
if
;~lDulZdr- ~
a
(x.D;OlDul2dr.
We infer the following nonexistence result of nontrivial solutions for problem (,). THEOREM 1.3. Let fl be a smooth bounded domain of [Rn. Let us suppose that there exists p ~ ]2, 2n/(n - 2)[ (if n _> 3, otherwise it suffices to require p > 2) such that the functions g e C°(R, JR) and A e c l ( f l , [R+) verify the following conditions (g)
(4)
t
tg(t)
g(O dr _< - j0 P
v t ~ [R;
Concentration phenomena
1015
Then, problem (.) has no positive solution, if g(t) > 0 for t > 0 and t) is strictly starshaped with respect to 0 (i.e. (x" v) > 0 on the boundary of f~). If the conditions (g) and (4) hold and, moreover, at least one of them is strictly verified, that is I i g ( r ) d r < - - tg(t) P
¥t#O
or
4 / [ 2 n + 2p -np[7 k P J
(x-D2)>
a.e. in f~,
then problem (,) has only the solution u - O, if ~ is starshaped with respect to 0 (i.e. (x. v) _> 0 on Of~).
Proof. Corollary 1.2 implies that
l
2IDu[2(x • v)da <_ -n
1
2 an
P
I
f
ug(u) dx + ~2 - n n 2lDu[ 2 d x -
n
II
~
n ( x ' D 2 ) l D u l 2dx.
Since u solves problem (,), one can easily verify that
I ug(u) d x = i n and so 1
2 oa
21OulZ(x • v ) d a <_ -n
P
a
2[DulZdx + ~2
-
n
n 2[Dul 2 dx - ~ n ( x . D~.)IDul 2 d x
because condition 0-) is fulfilled. Therefore, we have ½Ioa~.lDul2(x • v)da = 0, that is 2Du = 0 on 0t) if ~ is strictly starshaped with respect to 0; so Ion 2(Du • v) d e = 0. Since u is a solution of problem (.), it follows that In g(u) dx = 0. We infer that, if g(t) > 0 for t > 0, then there exists no positive solution. Moreover, if f~ is starshaped with respect to 0 and at least one of the conditions (g) or 0-) is strictly verified, then problem (.) has only the solution u -= 0: indeed, we should have in the preceding relations 1
f
2 on or
21Oul2(x • v ) a x < -n
P
l
u g ( u ) d x + -2- y- - n
n
l
1f
. 21DulZdx _ -2
n
(x. D2)[DuIZd x
1016
D. PASSASEO
In all the cases we should have the contradiction
[
±2
2]Dul2(x • v)dtr < 0.
J ofl Finally, notice that u --- 0 is a solution of the equation div(ADu) + g(u) = 0, because condition (g) implies that g(0) = (1/p)g(O) with p ;~ 1, that is g(0) = 0. So u - 0 is the unique solution o f problem (.) if at least one o f the conditions (g) or (2) is strictly fulfilled and t) is starshaped with respect to 0. •
Example 1.4. Let t) be a smooth bounded domain of R n, strictly starshaped with respect to 0; let 2(x) = Ixl and g(t) = tltl p-2 with p e ]2, 2n/(n - 2)[ (if n _> 3, otherwise it suffices p>2). Then theorem 1.3 implies that problem (,) has no positive solution for > (2n + 2p - np)/p (notice that (2n + 2p - np)/p > 0 for p ~ ]2, 2n/(n - 2)[) and has only the solution u -= 0 for ~ > (2n + 2p - np)/p. 2. S U F F I C I E N T C O N D I T I O N S FOR T H E E X I S T E N C E OF S O L U T I O N S
In this section we obtain a compactness property (see proposition 2.1) which gives a condition on the function 2 sufficient to guarantee existence and multiplicity results for problem (.), as in the case of uniformly elliptic problems (see theorem 2.2 and remark 2.3). Notice that the conditions required on 2 in proposition 2.1 and in theorem 2.2 are optimal, in the sense specified in remark 2.4. Together with the nonexistence results exposed in the preceding section, the results here stated describe completely the situation for problem (.). PROPOSITION 2.1. Let f~ be a bounded domain of ~n with n _ 3, p e ]2, 2n/(n - 2)[ and 2 e L®(f~) be a function almost everywhere positive in f~. Let us assume that there exists q>np/(2n+2p-np) such that l / 2 e L q ( ~ ) (notice that 2 n + 2 p n p > O and np/(2n + 2p - n p ) > 1 for p e ]2, 2n/(n - 2)[). Then, for every c > 0, the set
Ec = Iu E HI'2(~): t'n 2(x)IDuI2dx <-cl is relatively compact in LP(t)).
Proof. It suffices to prove that the set Ec is bounded in Hl'r(f~) for a suitable r e ]1, n[ verifying p < nr/(n - r), in such a way that Hd'r(~)) is embedded with compactness in LP(~). We can choose r = 2q/(1 + q), because the condition n r / ( n - r ) > p is fulfilled if q > np/(2n + 2p - n p ) (notice that 1 < r < 2 < n in our assumptions). So, using the Holder inequality, we have for every u e Ec f
fl lDulrdx
= f 2q/(l+q)[Dul2q/(l+q) Aq/O1 +q) dx fl
(l
[ II111
~q/(l+q)l/~
] q/(q+ 1)
1 ~l/(q+l)
Concentration phenomena
1017
The preceding compactness property allows to apply well-known topological methods of the calculus of variations to obtain existence and multiplicity of solutions for problem (.). One can state, for example, the following theorem. THEOREM 2.2. Let ~ be a bounded domain of IR" with n _> 3, p e ]2, 2 n / ( n - 2)[, and 2 e L®(f~) be a function almost everywhere positive in f2, such that 1/2 e Lq(~) with
q > np/(2n + 2p - n p ) . Then the problem
I divO.(x)Du(x)) + lulp-Eu = 0 u=Oona~,u~O
in f~ in
has infinitely many (weak) solutions, and at least one of them is positive.
Proof. Solving the problem considered is equivalent to looking for the critical points of the functional
f(u) : fa AlDulZdx constrained on the manifold
iu 2,o,
1
In fact, a function fi which is a critical point for f on Vp is a weak solution of the equation div0.Dfi) + ul lp-2
= 0
with p = f ( f i ) > 0; hence, the function u =/~l/(p-2)t2 solves the considered problem (conversely, if u is a solution of the problem, then the function t2 = u/llullLo( ) is a critical point for f on the constraint Vp). Since f is an even functional whose sublevels verify the compactness property described in proposition 2.1, well-known topological methods from the critical points theory (Krasnoselskii's genus, the Lyusternik-Schnirelman category for even functionals) enable one to state that the functional fconstrained on Vp has infinitely many pairs (u, -u) of constrained critical points. In particular, the preceding compactness property guarantees the existence of the minimum of f on Vp; since the minimum is achieved in a nonnegative function fi, the corresponding solution will be positive by the maximum principle. •
Remark 2.3. Analogous existence and multiplicity results for the solutions of problem (,) can be stated in the case that
....
tg(t)
0 < n m m~555~-- ~ lira sup and 1/2 ~ L q with q > np/(2n + 2p - np).
tg(t)
< +oo
1018
D. PASSASEO
Notice that finding nontrivial solutions of problem (.) is equivalent to looking for critical points of the functional f constrained on the manifold Vp =
u e H~'2(f]):
dx J~
g(r) dr = 1 J0
when g is a homogeneous function, while for the existence of infinitely many solutions it is very important that g is an odd function.
Remark 2.4. The existence and nonexistence results stated in theorems (1.3) and (2.2) are complementary and the conditions required in these two theorems are optimal in the sense that if, for example, A(x) = Ixl ° and g(t) = tltl p-2, then at least one of them can be applied: if a < (2n + 2p - np)/p, then theorem 2.2 guarantees the existence of solutions of problem (.) independently of the shape of O; on the contrary, if a _> (2n + 2p - np)/p and ~ is a bounded domain starshaped with respect to 0, then theorem 1.3 excludes that problem (,) can have nontrivial solutions. 3. M U L T I P L I C I T Y OF P O S I T I V E S O L U T I O N S
In this section we study the effect of the function 2 on the number of positive solutions of uniformly elliptic problems which approximate degenerate elliptic problems. We consider a family (2,)e > 0 of positive functions and we show that the solution of the corresponding problems P~, as e ~ 0 ÷, tends to "retire itself" in the degeneration set (that is the subset of O where 2~ --, 0), leaving the subset where 2, is uniformly positive. If the degeneration set is constituted by k connected components (k > 1), one can prove the existence of at least k + 1 distinct positive solutions of problems P, for e > 0 sufficiently small. In theorem 3.2 we prove this result only in a simple case: when the degeneration set is constituted by k balls, two by two disjoined (so we consider some simplified assumptions); moreover, we describe the qualitative properties of the solutions and their behaviour as e ~ 0 ÷. In remark 3.3 we point out some possible extensions of theorem 3.2, which can be obtained using analogous methods. The general result, concerning the case where the degeneration set has a more general shape, will be proved in a paper in preparation. We shall premise some notations and introduce the "barycentre" function fl(u), which has been a very useful tool for stating the existence and multiplicity results of positive solutions for uniformly elliptic problems with critical and subcritical Sobolev exponents, where analogous concentration phenomena occur (see [12, 21-26]).
Notation 3.1. Let B(x, r) = [y ~ [R": ly - x[ < r} for every x ~ IR" and r > 0. For every p e ]2, 2n/(n - 2)[ put pp = m i n i ( L JB(O, 1)
whereu + = uvO.
IDul2dx:ueHd'2(B(O, 1)), I
[ulPdx = 11 , B(O, 1)
Concentration phenomena
1019
Notice that ~÷ is a C2-submanifold of Hd'E(f~) with codimension 1. Let fl: ~+ --' R n be the function defined in the following way:
B(u)
= iJ t2 X ( U + ( X ) ~ ° dx.
THEOREM 3.2. Let t) be a smooth bounded domain of ~n, p e ]2, 2 n / ( n - 2[, x~ . . . . . xk in ~ and rl . . . . . r k in R ÷ be such that the spheres B(Xl, rl) . . . . . B(Xk, rk) (k > 1) are two by two disjoined and all included in ~. For every e > 0, let Aee C~(~, ~+) be a positive function such that: (a) lira inf(1/e infAe) > 0; ~--*0 +
f~
(b) 2~(x) = ,F.rj(2n+2p-np)/p V x ~_ B ( x j , rfl V j = 1 . . . . . k; (c) if P~. . . . . rk satisfy ~ > rj ¥ j = 1. . . . . k, then it results that lim x e t 2 / j yki B ( x j , Pj)1 > O. ,-,oinf + infIA,(x): (. Under these assumptions, there exists g > 0 such that v e e ]0, g[ the problem
P~
f div(,~ (x)Du) + u v - 1 = 0
in f2
(u
in f~
-- 0 o n a f t , u > 0
has at least k + 1 distinct solutions ue,~ . . . . . U e , ( k + l ) . Moreover, these solutions verify the following properties: (I) lim J o l D u , , j l 2 d x = 0 V j = 1 . . . . . (k + 1); ~--,0 +
(II) there exist ?~ . . . . . rk, with ~ > rj V j = 1 . . . . . k, v e e ]0, g[ the solution u,,j minimizes the functional
such that ¥ j =
1. . . . . k and
u ~ I~ &~lDul2 dx in the set
u ~ Hd'2(n): Ilu+ll. = Ilu~,jll., lim JB/(U*'J)P d x ~-'°+ In (u*,j) u d x = 1
(III) (iv) (V)
1:i(2n+ 2p-np)/p JB
lim
~o+ /~p
Ilu.,jll~
=/zp
v j = 1. . . . . k;
Vj = 1. . . . . k,
(see notation 3.1);
l Jn A, IDu,,k+tl2dx 1 Jn A, IDu,,k+I[ 2 d x <- 2tp-2)/Plap. < lim sup -< lim inf e-,o + e []ue,k+lllp 2 ~o+ ~ Ilu.,k+~ll~
1020
D. PASSASEO
Proof. Solving problem P, is equivalent to finding the constrained critical points u of the functional f,(u) = la ~ Ae(x)lDul2 dx on the constraint Vp+ (see notation 3.1). Indeed, a function u which is a constrained critical point for f~ on Vp+ is a weak solution of the equation div(A~Du) + lae(u+)a-I = O, for a suitable multiplier g, • JR. It follows that u _> 0 and /z~ = eft(u). Therefore, [ef~(u)]l/(a-2)u is a solution of P, (and it is strictly positive by the maximum principle). Let f~ . . . . . fk satisfy ~ > r i V i = 1. . . . . k; we prove that (u): u • Vp+, fl(u) • 1,30B(xj, ~)
lim inf inf e~O+
> ap:
(2)
j = 1
by contradiction, suppose that there exists a sequence of positive numbers (ei)i -~ 0 and a sequence of functions (ui)i in Vp+ such that k
B(ui) • ~ OB(xj, fj)
Vi • N
j=l
and
- ~p.
limfei(ui)
i--* oo
(3)
Since lim 1/e i inf2,i > 0, we infer that the sequence (ui)i is bounded in HoL2(O); hence, there i~oo
fl
exists a subsequence of (ui)i (which we shall denote again by (ui)i) converging to a function u • HI'E(D) weakly in H~'E(f~), in LP(D) and almost everywhere in D. Moreover, since for every rl ..... /~k such that ~ > I) v j = 1 . . . . . k, we have ;t.,(x): x • f~
lim inf inf i--* ao
(.•i
/ [..Jk B(xj, ~) 1 = +oo, j= 1
it must be k
vx • ~
u(x) = o
U O(xj, rj). j=l
Since/~(u) • [.Jjk = 1 #B(xj, #), there exists at least two integers s, t • {1. . . . . k} such that
f
(u+~dx
>
0
and
B(Xs , rs)
f
|
(u+) p dx > O.
,J B(xt , rt)
Hence, we have k
~p < ~ rJ2~+2p-~p)/p j= 1
IDul 2 <- f~,(u) B(xj,rj)
W i e N,
Concentration phenomena
1021
which implies k
up < E r:z"+2p-"p)/p j= I
IOul 2 dx B(xj, rj)
_< lim inf ~ i~oo
IOui I2 dx
r: 2"+2p-np)/p
j= l
B(xj,rj)
_< lim. fei(Ui) l~eo
in contradiction with (3). So (2) is proved. Now, let us verify that v e > 0 and v j = 1. . . . . k it results that inf[f~(u): u ~ Vp+, fl(u) ~ B(xj, 0)l < Up:
(4)
in fact, let fi be a positive function which gives the minimum Up, that is fi e HI'2(B(0, 1)), >0
in B(0, 1),
( Ifilp (ix = 1, J B(O, 1)
l
IDtzl2dx = Up;
J a ( o , 1)
set o,-
So, it results that ~j e Vp÷, fl(t~j) e B(xj, respect to xj) and, moreover,
O)
f~(ftj) = Up
.
(indeed fl(t~j) = xj because ui is symmetric with v j = 1 . . . . . k,
which implies (4). From (2) and (4) we infer that, for e > 0 small enough and v j = 1. . . . . k, there exists a function t~,j minimizer of the functional f , in the set (u ~ v : : ,e(u) ~
B(xj, 0)1
(notice that t~,.j is a nonnegative function because t~,,j e Vp+ if and only if (~,j)+ e Vp+ and -+
L(u.,j) <- f~(f~.,i)). Hence, it is easy to verify that the corresponding solution u~,j = [ef,(ft, d)]l/(p-2)~,,j satisfies property (II). Moreover, we have evidently
\ Ilu~,5 I1# -< UpLet us prove that there exists another constrained critical point ft,,k+ l for fe on Vp+: let y: [0, 1] ~ Vp+ be the continuous curve joining the positive functions t~ and fi2, defined in the following way z~ 1 + (1 - "t)/~2 ~,(r) = [Ira~ + (1 - r)a2ll."
1022
D. PASSASEO
It is easy to verify that max{f~ o y(~'): z ~ [0, 1]} _< 2(P-2)/Pll p
V ~ > 0.
(6)
Therefore, the functions t~1 and u2 cannot be joined by a continuous curve included in the sublevel [/'g ~ V p + : f e ( u ) --~ Up},
because this sublevel does not meet the set [u e Vp+:/~(u) ~
OB(x,,
rl)}
for e > 0 small enough (see (2)), while they are joined in the sublevel [U e Vp+:fe(u) _< 2 ( p - 2 ) % 1 ,
by the continuous curve ~, (see (6)). Moreover, since inf2~ > 0 and p e ]2, 2n/(n - 2)[, the functional f~ constrained on the a manifold Vp+ verifies the well known Palais-Smale compactness condition. Hence, by the " m o u n t a i n pass" theorem of Ambrosetti-Rabinowitz, there exists a critical value forf~, constrained on Vp+, in ]/19,2(p-2)/plzp]. If we call fi~,k+~ the corresponding critical point, the function [g-"fe (/~E, k+ 1)] l/(p-2)l-le, k + 1
lle:, k+ 1 :
is a solution of problem P~, distinct from the previous ones. Let us prove (I): since lira supf~(fi~,j) <
v j = 1. . . . . k + 1,
+oo
C~0 +
lira inf 1 inf(A~) > 0, e~O +
~
fl
there exists a constant c > 0 such that, for e > 0 small enough and v j = 1. . . . . k + 1, we have
I IDu~'jl2dx= [ef~(fl~'J)]2/(p-2)lo < [eft(flt,j)]2/(P-2)cfe(~le,j)--* 0
as e ~ 0 +.
Property (II) follows evidently by the method used to obtain the solutions u~,i f o r j = 1 . . . . . k. Let us prove (III): since fi~,j = u~,j/[~a (ue,j) p dx] l/p, because ue,j is a positive function, we have
l
Btxj, rj)
(~,j)p dx <_ 1.
By contradiction, suppose that there exists an infinitesimal sequence o f positive numbers (ei)i such that lira
i
i~oo 3B(xj,rj)
(~,,,j)P dx
< 1.
(7)
Concentration phenomena
1023
Since f~(fte,,j) < I~p and lim inf 1 infO.D > O, ~--)0 +
~
t)
the sequence (u~i,j)i is bounded in H~'E(D). Therefore, (f%,j)i, or a subsequence, converges in H~'2(~) weakly, in L"(£)) and almost everywhere in D to a function fij. Since for every P1 . . . . , rk such that ~ > rj y j = 1 . . . . . k one has liminfinf_lA.i(x): x e ~ i~ ~ (.ei
B(xj, ~)
=
= +~,
it results k
~j=0
vx ~~
O B(xj, rj). j=l
Moreover, by (7) we have JB(xj,O (fiJ)P dx < 1, which implies
ID~jl 2 dx
~p < E r}2~+2p-'p)/" s= 1
B(xs,r,)
and so ~lp <
E
< --
[DYzj[ 2 d x
r(s 2 n + 2 p - n p ) / p
s= 1
B(xs,rs)
lim inf ~ -$ r(2n+2p-np)/P i--*oo
j= 1
[Dfl,i,j]2 dx B(xs,rs)
_< lim inff~,(fi,, j) in contradiction with the fact that lim inff~i(~,/) _
V j = l, ..., k.
Assume, by contradiction, that there exists an infinitesimal sequence of positive numbers (ei)i such that Jim
r(2"+2p-np)/p
l ~ ~o
t"
[Dflti,J 12 d x < t i p .
(8)
J B(xj , rj)
Then one can prove that the sequence (ue~,j)i is bounded in /_/~,z(~) and there exists a subsequence converging in LP(~) and almost everywhere in ~ to a function ~j which is zero in ~ \ B ( x j , rj). It follows that
llp < r>2n+2P-nP)/PI
IDflj[2 dx < lim r>2n+2p-nP)/Pf B(Xj, rj)
in contradiction with (8).
i ~ oo
,D~le,,j]2dx B(Xj, rj)
1024
D. PASSASEO
Property (V) is a simple consequence of the fact that
fe(tle,(k + l)) ~ ]flU, 2(P- 2)/p/lp] for every e > 0 small enough. Notice that this means, in particular, that the function /~e,(k+l) converges, as e ~ 0 ÷, to a nonnegative function t~tk+l) which differs from zero at most in two of the spheres B(xj, ri), j = 1 . . . . . k. •
Remark 3.3. The preceding theorem describes the effect of the concentration phenomena in uniformly elliptic problems, which approximate degenerate elliptic problems, in a simplified situation. It is clear that simple modifications of the proof enable one to obtain an analogous result in the case that, instead of the spheres B(xj, rfl, the connected components of the degeneration set are more general subsets Ci, with convex hulls two by two disjoined. However, assumptions of this type, which have an important role in the proof of theorem 3.2, are not necessary: in fact an analogous multiplicity result (proved in a paper in preparation) can be stated also when the degeneration set has a very general shape. REFERENCES 1. AMBROSETTI A. & RABINOWITZ P., Dual variational methods in critical point theory and applications, J. funct. Analysis 14, 349-381 (1973). 2. RABINOWITZ P., Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. math. J. 23, 729-754 (1974). 3. FABES E., KENIG C. & SERAPIONI R., The local regularity of solutions of degenerate elliptic equations, Communs partial, diff. Eqns 7, 77-116 (1982). 4. MURTHY M. K. V. & STAMPACCHIA G., Boundary value problems for some degenerate elliptic operators, Annali. Mat. pura appl. 80, 1-122 (1968) 5. STREDULINSKY E. W., Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Math. Vol. 107. Springer, Berlin (1980). 6. POHOZAEV S. I., Eigenfunctions of the equation Au + 2f(u) = O, Soviet. Math. Dokl. 6, 1408-1411 (1965). 7. BREZIS H. & NIRENBERG L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communs pure appl. Math. 36, 437-477 (1983). 8. BREZIS H. & NIRENBERG L., A minimization problem with critical exponent and non zero data, in Symmetry in Nature, pp. 129-140. Scuola Norm. Sup., Pisa (1989). 9. BREZIS H., Elliptic equations with limiting Sobolez exponents--the impact of topology, in Proceedings 50th Anniv. Courant lnst., Communs pure appl. Math. 39, 17-39 (1986). 10. LIONS P. L., The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana 1, 145-201; 45-121 (1985). 11. STRUWE M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities Math. Z. 187, 511-517 (1984). 12. PASSASEO D., Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math. 65, 147-166 (1989). 13. PASSASEO D., Nonexistence results for elliptic problems with supercriticai nonlinearity in nontriviai domains, J. funct. Analysis (to appear). 14. PASSASEO D., Risultati di non esistenza per equazioni ellittiche con nonlinearit~ sopraeritica. Preprint, Dip. Mat., Pisa (1992). 15. PASSASEO D., Existence e molteplicit~t di soluzioni positive per equazioni ellittiche con nonlinearit/l sopracritica. Preprint No. 619, Dip. Mat., Pisa (1992). 16. PASSASEO D., Esistenza e molteplicit/l di soluzioni nodali per equazioni ellittiche con esponente sopracritico in aperti contrattili. Preprint, Dip. Mat., Pisa (1992). 17. BAHRI A. & CORON J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Communs pure appl. Math. 41, 253-294 (1988). 18. CORON J. M., Topologie et cas limite des injections de Sobolev, Cr. hebd. S~anc. Acad. Sci. Paris Ser. 1 299, 209-212 (1984). 19. DANCER E. N., A note on an equation with critical exponent, Bull. London math. Soc. 20, 600-602 (1988).
Concentration phenomena
1025
20. DING W., Positive solutions of Au(n + 2)/(n - 2) = 0 on contractible domains, J. Partial diff. Eqns 2(4), 83-88 (1989). 21. PASSASEO D., Problemi ellittici con esponente critico. Forma del dominio e molteplicith di soluzioni positive. Preprint No. 564, Dip. Mat. Pisa (1990) 22. PASSASEO D., Su alcune successioni di soluzioni positive di problemi ellittici con esponente critico, Rend. Mat. Acc. Lincei 3, 15-21 (1992). 23. BENCI V. & CERAMI G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Archs ration. Mech. Analysis. 114, 79-93 (1991). 24. BENCI V., CERAMI G. & PASSASEO D., On the number of positive solutions of some nonlinear elliptic problems, in Nonlinear Analysis. A Tribute in Honour o f G. Prodi (Edited by A. AMBROSETTI and A. MARINO), pp. 93-107. Quaderni Scuola Norm. Sup., Pisa (1991). 25. CERAMI G. & PASSASEO D., Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with rich topology, Nonlinear Analysis 18, 109-119 (1992). 26. CERAMI G. & PASSASEO D., Existence and multiplicity results for semilinear elliptic Drichlet problems in exterior domains. Preprint, Dip. Mat., Pisa. 664 (1992).
Pergamon 0362-546X(94)00127-8
SOME CONCENTRATION DEGENERATE SEMILINEAR
PHENOMENA IN ELLIPTIC PROBLEMS
DONATO PASSASEO Dipartimento di Matematica, Universitfi di Pisa, Via Buonarroti 2, 56127-Pisa, Italy
(Received 16 July 1993; received f o r publication 31 May 1994) Key words and phrases: Nonlinear Dirichlet problems, degenerate equations, concentration p h e n o m e n a and nonexistence results, multiplicity of positive solutions.
0. I N T R O D U C T I O N
Let ~ be a smooth bounded domain of R ~, )t ~ L®(~) be a function almost everywhere positive in D and g: ~ ~ ~ be a continuous function such that g(0) = 0 and as t . ~ +_oo,
tg(t) - [ t f
w i t h p > 2 a n d p < 2 n / ( n - 2) if n > 3. In this paper we study the existence and the multiplicity of nontrivial solutions u (i.e. u ~ 0 in ~) for the problem
(*)
I div(;tDu) + g(u) = 0
in f~
(. u = 0
on 0t~.
Well-known results guarantee existence and multiplicity of (positive or nodal) solutions for problem (,) in the case that infA > 0 (see, for example, [1, 2]). u
On the contrary, several problems arise when the equation is degenerate, that is inf 2 = 0: in this case the classical topological methods of the calculus of variations cannot be applied directly and the existence and multiplicity results, which hold when i n f 2 > 0, cannot be extended to the degenerate case. Many papers have been devoted to the study of several questions concerning degenerate elliptic problems (see, for example, [3-5] and the references therein). In this paper we study some concentration phenomena for the solutions of problem (.), which recall some situations analogous to those occurring in elliptic problems with critical or supercritical Sobolev exponents (see [6-16]). We show that, as in those problems, existence, nonexistence and multiplicity of nontrivial solutions for problem (.) are closely related to these phenomena. In our case, such concentration phenomena are due not to the Sobolev exponent 2 n / ( n - 2), but to the degenerate character of the differential equation, as shown by the following example: let ~,(x) = Ix[" with c~ > 0 and suppose that 0 e ~ and g(t) tltlP-2; as is well known, for a function g of this type, the nontrivial solutions of problem (,) correspond to the critical points of the functional =
f ( u ) = I ~.(x)[Du[ 2 dx J fi lOll
1012
D. PASSASEO
constrained on the manifold
I If o~ < (2n + 2p - np)/p, then there exists the minimum o f f on ~ and so problem (.) has a (positive) solution (see theorem 2.2); on the contrary, if ce > (2n + 2p - np)/p, then this minimum does not exist because i n f f = 0, as one can easily verify (it suffices to put
vp ¢,(x) = (o(x/e) with ¢~ e C~(B(O, 1)) and to remark that lim f(~,/]lgo, l],) = 0). e~0 +
Moreover, the results proved in this paper (see theorem 1.3) show that, if f~ is starshaped with respect to 0 and ct _ (2n + 2p - np)/p, not only does the minimum o f f on the constraint Vo not exist, but there is not even a constrained critical point. On the contrary, if 2 is infinitesimal of order small enough (in the sense that 1/2 e if(g)) for a suitable q > 1), then one can prove a compactness property (see proposition 2. l) which enables one to obtain general existence and multiplicity results as in the case inf A > 0. n In the problems with critical and supercritical exponent considered in [6, 7, 9, 12-22], as well as in the ones studied in [23-26], some concentration phenomena like these enable one to relate the number of positive solutions to the shape of the domain f~; in an analogous way, it is possible to relate the multiplicity of positive solutions of problem (,) to the properties of the function 3., as shown by the results stated in Section 3 (see theorem 3.2 and remark 3.3). The paper is organized as follows. In Section 1, we prove an identity (like the Pohozaev identity [6]) which gives some conditions on functions ~. and g, sufficient to exclude the existence of nontrivial solutions for problem (.) in starshaped domains; in Section 2, on the contrary, we give sufficient conditions for the existence and the multiplicity of nontrivial solutions for problem (.); in Section 3, we study the effect of the properties of function 2 on the number o f positive solutions of problem (,): we consider a family of functions (~-,),>o, such that inf 2~ > 0
ve > 0
fl
and
lim inf 2~ = O. ,~0 +
fl
For these uniformly elliptic problems P~, which approximate degenerate elliptic problems, we find some solutions which, as e -~ 0 +, tend to "retire itself" in the subset of ~ where ~-e ~ 0; hence, if the subset where ~ ~ 0 is constituted by k connected components, we prove the existence o f at least k + 1 positive solutions for e > 0 small enough, and we describe their qualitative properties and the asymptotic behaviour as e ~ 0 + (see theorem 3.2 and remark 3.3). Let us note that the nonexistence results of Section 1 and the existence results given in Section 2 are complementary and describe completely the situation: indeed, the example considered above (2(x) = Ixl% g(t) = t It [p-2, n starshaped with respect to 0) satisfies the assumptions of theorem 1.3 or those of theorem 2.2; hence the assumptions considered in these two theorems are optimal. On the contrary, theorem 3.2 is stated in a simplified form; some of its possible generalizations have been pointed out in remark 3.3. 1. N O N E X I S T E N C E
RESULTS
In this section we prove that the solutions of problem (.) verify a general identity which, in a particular case, recalls the Pohozaev identity (see [6]), and is very important in studying
Concentration phenomena
1013
elliptic problems with critical and supercritical Sobolev exponents (see lemma 1.1 and corollary 1.2). This identity gives a condition on functions ). and g, which is sufficient to exclude the existence of nontrivial solutions in starshaped domains D (see theorem 1.3). LEMMA 1.1. Let D be a smooth bounded domain of ~ ' , ;t be a smooth real function almost everywhere positive in D, g: R --, ~ be a continuous function and u be a solution of the problem
l div(;t(x)Du(x)) + g(u(x)) =
0
u = 0
in D on OD.
Then, for every vector field v e CI(~, ~'), it results in
1 I ~ Di2vilDul2dx,
2 ~i=1
where v denotes the outward normal to the boundary of t) and
Proof. For every w e
G(t) -- I[ g(r) dr.
cl(~r~, [~) one has
f div(2Du)wdx+In g(u)w dx = O, that is fan'~w(Du'v)dtr-f9
2DuDwdx + 19 g(u)wdx : O.
In particular, if for a fixed v e C1(~, ~ ' ) we set
Du = (Du. v)v on the boundary of £))
w = (Du'v),
then we have (since
lan A'Dul2(v " v) da = fg "~j~IDiUDi( i~l DiUvi) dx - t'g g(u)( i~=l DiUvi) dx, that is
f 09 XlDul2(v"v) da= I 2fl i,j~=1DjuDijuvidx+ f fl i,j=l DjuDiuDjvi dx - f9 g(u) i=l ~ Diuvidx and so l
o. ;tlDul2(v"
~Di(lDul2)vidx+ f 9 i,j=l ADivjDiuDju dx
v) d a = ~I I 9 A i = ,
-- ig i=~ l Di[G(u)I vidx'
(1)
1014
D. PASSASEO
which implies
12 on ;qDul2tv • v) d a = - 2 ,1j f l i = l~ Di()tVi)lDu[ 2 dx +
2DiujDiuDju dx
+ ( G(u) div v d r , J that is
1 2 oa
~lOuff(v'v) d a = - ~ 1 .i=~ ~, Dido, IDul 2 dr - ~
fl i , j = 1
[~i=l
~Div, IDul2 dr
fl
= fa [ G(u) - 2 ~D-~] div(v)dx + laA
i,j=l
DivjDiuDju dx
1 I IDul2 ~ Di;tvidx.
2 a
i=1
In particular, if in the previous lemma we set v(x) = x, we obtain for the solutions of problem (,) a relation analogous to the Pohozaev identity (see [6]). COROLLARY1.2. Let ~, ~., g verify the same assumptions of lemma 1.1. Then, for every solution of problem I div(2Du) +
g(u) = 0
in g)
u = 0
on O~
we have (with the same notations of lemma 1.1)
1
2 on
;~lDul2(x • v)da
-- n
S
n
G(u)dx +
a
if
;~lDulZdr- ~
a
(x.D;OlDul2dr.
We infer the following nonexistence result of nontrivial solutions for problem (,). THEOREM 1.3. Let fl be a smooth bounded domain of [Rn. Let us suppose that there exists p ~ ]2, 2n/(n - 2)[ (if n _> 3, otherwise it suffices to require p > 2) such that the functions g e C°(R, JR) and A e c l ( f l , [R+) verify the following conditions (g)
(4)
t
tg(t)
g(O dr _< - j0 P
v t ~ [R;
Concentration phenomena
1015
Then, problem (.) has no positive solution, if g(t) > 0 for t > 0 and t) is strictly starshaped with respect to 0 (i.e. (x" v) > 0 on the boundary of f~). If the conditions (g) and (4) hold and, moreover, at least one of them is strictly verified, that is I i g ( r ) d r < - - tg(t) P
¥t#O
or
4 / [ 2 n + 2p -np[7 k P J
(x-D2)>
a.e. in f~,
then problem (,) has only the solution u - O, if ~ is starshaped with respect to 0 (i.e. (x. v) _> 0 on Of~).
Proof. Corollary 1.2 implies that
l
2IDu[2(x • v)da <_ -n
1
2 an
P
I
f
ug(u) dx + ~2 - n n 2lDu[ 2 d x -
n
II
~
n ( x ' D 2 ) l D u l 2dx.
Since u solves problem (,), one can easily verify that
I ug(u) d x = i n and so 1
2 oa
21OulZ(x • v ) d a <_ -n
P
a
2[DulZdx + ~2
-
n
n 2[Dul 2 dx - ~ n ( x . D~.)IDul 2 d x
because condition 0-) is fulfilled. Therefore, we have ½Ioa~.lDul2(x • v)da = 0, that is 2Du = 0 on 0t) if ~ is strictly starshaped with respect to 0; so Ion 2(Du • v) d e = 0. Since u is a solution of problem (.), it follows that In g(u) dx = 0. We infer that, if g(t) > 0 for t > 0, then there exists no positive solution. Moreover, if f~ is starshaped with respect to 0 and at least one of the conditions (g) or 0-) is strictly verified, then problem (.) has only the solution u -= 0: indeed, we should have in the preceding relations 1
f
2 on or
21Oul2(x • v ) a x < -n
P
l
u g ( u ) d x + -2- y- - n
n
l
1f
. 21DulZdx _ -2
n
(x. D2)[DuIZd x
1016
D. PASSASEO
In all the cases we should have the contradiction
[
±2
2]Dul2(x • v)dtr < 0.
J ofl Finally, notice that u --- 0 is a solution of the equation div(ADu) + g(u) = 0, because condition (g) implies that g(0) = (1/p)g(O) with p ;~ 1, that is g(0) = 0. So u - 0 is the unique solution o f problem (.) if at least one o f the conditions (g) or (2) is strictly fulfilled and t) is starshaped with respect to 0. •
Example 1.4. Let t) be a smooth bounded domain of R n, strictly starshaped with respect to 0; let 2(x) = Ixl and g(t) = tltl p-2 with p e ]2, 2n/(n - 2)[ (if n _> 3, otherwise it suffices p>2). Then theorem 1.3 implies that problem (,) has no positive solution for > (2n + 2p - np)/p (notice that (2n + 2p - np)/p > 0 for p ~ ]2, 2n/(n - 2)[) and has only the solution u -= 0 for ~ > (2n + 2p - np)/p. 2. S U F F I C I E N T C O N D I T I O N S FOR T H E E X I S T E N C E OF S O L U T I O N S
In this section we obtain a compactness property (see proposition 2.1) which gives a condition on the function 2 sufficient to guarantee existence and multiplicity results for problem (.), as in the case of uniformly elliptic problems (see theorem 2.2 and remark 2.3). Notice that the conditions required on 2 in proposition 2.1 and in theorem 2.2 are optimal, in the sense specified in remark 2.4. Together with the nonexistence results exposed in the preceding section, the results here stated describe completely the situation for problem (.). PROPOSITION 2.1. Let f~ be a bounded domain of ~n with n _ 3, p e ]2, 2n/(n - 2)[ and 2 e L®(f~) be a function almost everywhere positive in f~. Let us assume that there exists q>np/(2n+2p-np) such that l / 2 e L q ( ~ ) (notice that 2 n + 2 p n p > O and np/(2n + 2p - n p ) > 1 for p e ]2, 2n/(n - 2)[). Then, for every c > 0, the set
Ec = Iu E HI'2(~): t'n 2(x)IDuI2dx <-cl is relatively compact in LP(t)).
Proof. It suffices to prove that the set Ec is bounded in Hl'r(f~) for a suitable r e ]1, n[ verifying p < nr/(n - r), in such a way that Hd'r(~)) is embedded with compactness in LP(~). We can choose r = 2q/(1 + q), because the condition n r / ( n - r ) > p is fulfilled if q > np/(2n + 2p - n p ) (notice that 1 < r < 2 < n in our assumptions). So, using the Holder inequality, we have for every u e Ec f
fl lDulrdx
= f 2q/(l+q)[Dul2q/(l+q) Aq/O1 +q) dx fl
(l
[ II111
~q/(l+q)l/~
] q/(q+ 1)
1 ~l/(q+l)
Concentration phenomena
1017
The preceding compactness property allows to apply well-known topological methods of the calculus of variations to obtain existence and multiplicity of solutions for problem (.). One can state, for example, the following theorem. THEOREM 2.2. Let ~ be a bounded domain of IR" with n _> 3, p e ]2, 2 n / ( n - 2)[, and 2 e L®(f~) be a function almost everywhere positive in f2, such that 1/2 e Lq(~) with
q > np/(2n + 2p - n p ) . Then the problem
I divO.(x)Du(x)) + lulp-Eu = 0 u=Oona~,u~O
in f~ in
has infinitely many (weak) solutions, and at least one of them is positive.
Proof. Solving the problem considered is equivalent to looking for the critical points of the functional
f(u) : fa AlDulZdx constrained on the manifold
iu 2,o,
1
In fact, a function fi which is a critical point for f on Vp is a weak solution of the equation div0.Dfi) + ul lp-2
= 0
with p = f ( f i ) > 0; hence, the function u =/~l/(p-2)t2 solves the considered problem (conversely, if u is a solution of the problem, then the function t2 = u/llullLo( ) is a critical point for f on the constraint Vp). Since f is an even functional whose sublevels verify the compactness property described in proposition 2.1, well-known topological methods from the critical points theory (Krasnoselskii's genus, the Lyusternik-Schnirelman category for even functionals) enable one to state that the functional fconstrained on Vp has infinitely many pairs (u, -u) of constrained critical points. In particular, the preceding compactness property guarantees the existence of the minimum of f on Vp; since the minimum is achieved in a nonnegative function fi, the corresponding solution will be positive by the maximum principle. •
Remark 2.3. Analogous existence and multiplicity results for the solutions of problem (,) can be stated in the case that
....
tg(t)
0 < n m m~555~-- ~ lira sup and 1/2 ~ L q with q > np/(2n + 2p - np).
tg(t)
< +oo
1018
D. PASSASEO
Notice that finding nontrivial solutions of problem (.) is equivalent to looking for critical points of the functional f constrained on the manifold Vp =
u e H~'2(f]):
dx J~
g(r) dr = 1 J0
when g is a homogeneous function, while for the existence of infinitely many solutions it is very important that g is an odd function.
Remark 2.4. The existence and nonexistence results stated in theorems (1.3) and (2.2) are complementary and the conditions required in these two theorems are optimal in the sense that if, for example, A(x) = Ixl ° and g(t) = tltl p-2, then at least one of them can be applied: if a < (2n + 2p - np)/p, then theorem 2.2 guarantees the existence of solutions of problem (.) independently of the shape of O; on the contrary, if a _> (2n + 2p - np)/p and ~ is a bounded domain starshaped with respect to 0, then theorem 1.3 excludes that problem (,) can have nontrivial solutions. 3. M U L T I P L I C I T Y OF P O S I T I V E S O L U T I O N S
In this section we study the effect of the function 2 on the number of positive solutions of uniformly elliptic problems which approximate degenerate elliptic problems. We consider a family (2,)e > 0 of positive functions and we show that the solution of the corresponding problems P~, as e ~ 0 ÷, tends to "retire itself" in the degeneration set (that is the subset of O where 2~ --, 0), leaving the subset where 2, is uniformly positive. If the degeneration set is constituted by k connected components (k > 1), one can prove the existence of at least k + 1 distinct positive solutions of problems P, for e > 0 sufficiently small. In theorem 3.2 we prove this result only in a simple case: when the degeneration set is constituted by k balls, two by two disjoined (so we consider some simplified assumptions); moreover, we describe the qualitative properties of the solutions and their behaviour as e ~ 0 ÷. In remark 3.3 we point out some possible extensions of theorem 3.2, which can be obtained using analogous methods. The general result, concerning the case where the degeneration set has a more general shape, will be proved in a paper in preparation. We shall premise some notations and introduce the "barycentre" function fl(u), which has been a very useful tool for stating the existence and multiplicity results of positive solutions for uniformly elliptic problems with critical and subcritical Sobolev exponents, where analogous concentration phenomena occur (see [12, 21-26]).
Notation 3.1. Let B(x, r) = [y ~ [R": ly - x[ < r} for every x ~ IR" and r > 0. For every p e ]2, 2n/(n - 2)[ put pp = m i n i ( L JB(O, 1)
whereu + = uvO.
IDul2dx:ueHd'2(B(O, 1)), I
[ulPdx = 11 , B(O, 1)
Concentration phenomena
1019
Notice that ~÷ is a C2-submanifold of Hd'E(f~) with codimension 1. Let fl: ~+ --' R n be the function defined in the following way:
B(u)
= iJ t2 X ( U + ( X ) ~ ° dx.
THEOREM 3.2. Let t) be a smooth bounded domain of ~n, p e ]2, 2 n / ( n - 2[, x~ . . . . . xk in ~ and rl . . . . . r k in R ÷ be such that the spheres B(Xl, rl) . . . . . B(Xk, rk) (k > 1) are two by two disjoined and all included in ~. For every e > 0, let Aee C~(~, ~+) be a positive function such that: (a) lira inf(1/e infAe) > 0; ~--*0 +
f~
(b) 2~(x) = ,F.rj(2n+2p-np)/p V x ~_ B ( x j , rfl V j = 1 . . . . . k; (c) if P~. . . . . rk satisfy ~ > rj ¥ j = 1. . . . . k, then it results that lim x e t 2 / j yki B ( x j , Pj)1 > O. ,-,oinf + infIA,(x): (. Under these assumptions, there exists g > 0 such that v e e ]0, g[ the problem
P~
f div(,~ (x)Du) + u v - 1 = 0
in f2
(u
in f~
-- 0 o n a f t , u > 0
has at least k + 1 distinct solutions ue,~ . . . . . U e , ( k + l ) . Moreover, these solutions verify the following properties: (I) lim J o l D u , , j l 2 d x = 0 V j = 1 . . . . . (k + 1); ~--,0 +
(II) there exist ?~ . . . . . rk, with ~ > rj V j = 1 . . . . . k, v e e ]0, g[ the solution u,,j minimizes the functional
such that ¥ j =
1. . . . . k and
u ~ I~ &~lDul2 dx in the set
u ~ Hd'2(n): Ilu+ll. = Ilu~,jll., lim JB/(U*'J)P d x ~-'°+ In (u*,j) u d x = 1
(III) (iv) (V)
1:i(2n+ 2p-np)/p JB
lim
~o+ /~p
Ilu.,jll~
=/zp
v j = 1. . . . . k;
Vj = 1. . . . . k,
(see notation 3.1);
l Jn A, IDu,,k+tl2dx 1 Jn A, IDu,,k+I[ 2 d x <- 2tp-2)/Plap. < lim sup -< lim inf e-,o + e []ue,k+lllp 2 ~o+ ~ Ilu.,k+~ll~
1020
D. PASSASEO
Proof. Solving problem P, is equivalent to finding the constrained critical points u of the functional f,(u) = la ~ Ae(x)lDul2 dx on the constraint Vp+ (see notation 3.1). Indeed, a function u which is a constrained critical point for f~ on Vp+ is a weak solution of the equation div(A~Du) + lae(u+)a-I = O, for a suitable multiplier g, • JR. It follows that u _> 0 and /z~ = eft(u). Therefore, [ef~(u)]l/(a-2)u is a solution of P, (and it is strictly positive by the maximum principle). Let f~ . . . . . fk satisfy ~ > r i V i = 1. . . . . k; we prove that (u): u • Vp+, fl(u) • 1,30B(xj, ~)
lim inf inf e~O+
> ap:
(2)
j = 1
by contradiction, suppose that there exists a sequence of positive numbers (ei)i -~ 0 and a sequence of functions (ui)i in Vp+ such that k
B(ui) • ~ OB(xj, fj)
Vi • N
j=l
and
- ~p.
limfei(ui)
i--* oo
(3)
Since lim 1/e i inf2,i > 0, we infer that the sequence (ui)i is bounded in HoL2(O); hence, there i~oo
fl
exists a subsequence of (ui)i (which we shall denote again by (ui)i) converging to a function u • HI'E(D) weakly in H~'E(f~), in LP(D) and almost everywhere in D. Moreover, since for every rl ..... /~k such that ~ > I) v j = 1 . . . . . k, we have ;t.,(x): x • f~
lim inf inf i--* ao
(.•i
/ [..Jk B(xj, ~) 1 = +oo, j= 1
it must be k
vx • ~
u(x) = o
U O(xj, rj). j=l
Since/~(u) • [.Jjk = 1 #B(xj, #), there exists at least two integers s, t • {1. . . . . k} such that
f
(u+~dx
>
0
and
B(Xs , rs)
f
|
(u+) p dx > O.
,J B(xt , rt)
Hence, we have k
~p < ~ rJ2~+2p-~p)/p j= 1
IDul 2 <- f~,(u) B(xj,rj)
W i e N,
Concentration phenomena
1021
which implies k
up < E r:z"+2p-"p)/p j= I
IOul 2 dx B(xj, rj)
_< lim inf ~ i~oo
IOui I2 dx
r: 2"+2p-np)/p
j= l
B(xj,rj)
_< lim. fei(Ui) l~eo
in contradiction with (3). So (2) is proved. Now, let us verify that v e > 0 and v j = 1. . . . . k it results that inf[f~(u): u ~ Vp+, fl(u) ~ B(xj, 0)l < Up:
(4)
in fact, let fi be a positive function which gives the minimum Up, that is fi e HI'2(B(0, 1)), >0
in B(0, 1),
( Ifilp (ix = 1, J B(O, 1)
l
IDtzl2dx = Up;
J a ( o , 1)
set o,-
So, it results that ~j e Vp÷, fl(t~j) e B(xj, respect to xj) and, moreover,
O)
f~(ftj) = Up
.
(indeed fl(t~j) = xj because ui is symmetric with v j = 1 . . . . . k,
which implies (4). From (2) and (4) we infer that, for e > 0 small enough and v j = 1. . . . . k, there exists a function t~,j minimizer of the functional f , in the set (u ~ v : : ,e(u) ~
B(xj, 0)1
(notice that t~,.j is a nonnegative function because t~,,j e Vp+ if and only if (~,j)+ e Vp+ and -+
L(u.,j) <- f~(f~.,i)). Hence, it is easy to verify that the corresponding solution u~,j = [ef,(ft, d)]l/(p-2)~,,j satisfies property (II). Moreover, we have evidently
\ Ilu~,5 I1# -< UpLet us prove that there exists another constrained critical point ft,,k+ l for fe on Vp+: let y: [0, 1] ~ Vp+ be the continuous curve joining the positive functions t~ and fi2, defined in the following way z~ 1 + (1 - "t)/~2 ~,(r) = [Ira~ + (1 - r)a2ll."
1022
D. PASSASEO
It is easy to verify that max{f~ o y(~'): z ~ [0, 1]} _< 2(P-2)/Pll p
V ~ > 0.
(6)
Therefore, the functions t~1 and u2 cannot be joined by a continuous curve included in the sublevel [/'g ~ V p + : f e ( u ) --~ Up},
because this sublevel does not meet the set [u e Vp+:/~(u) ~
OB(x,,
rl)}
for e > 0 small enough (see (2)), while they are joined in the sublevel [U e Vp+:fe(u) _< 2 ( p - 2 ) % 1 ,
by the continuous curve ~, (see (6)). Moreover, since inf2~ > 0 and p e ]2, 2n/(n - 2)[, the functional f~ constrained on the a manifold Vp+ verifies the well known Palais-Smale compactness condition. Hence, by the " m o u n t a i n pass" theorem of Ambrosetti-Rabinowitz, there exists a critical value forf~, constrained on Vp+, in ]/19,2(p-2)/plzp]. If we call fi~,k+~ the corresponding critical point, the function [g-"fe (/~E, k+ 1)] l/(p-2)l-le, k + 1
lle:, k+ 1 :
is a solution of problem P~, distinct from the previous ones. Let us prove (I): since lira supf~(fi~,j) <
v j = 1. . . . . k + 1,
+oo
C~0 +
lira inf 1 inf(A~) > 0, e~O +
~
fl
there exists a constant c > 0 such that, for e > 0 small enough and v j = 1. . . . . k + 1, we have
I IDu~'jl2dx= [ef~(fl~'J)]2/(p-2)lo < [eft(flt,j)]2/(P-2)cfe(~le,j)--* 0
as e ~ 0 +.
Property (II) follows evidently by the method used to obtain the solutions u~,i f o r j = 1 . . . . . k. Let us prove (III): since fi~,j = u~,j/[~a (ue,j) p dx] l/p, because ue,j is a positive function, we have
l
Btxj, rj)
(~,j)p dx <_ 1.
By contradiction, suppose that there exists an infinitesimal sequence o f positive numbers (ei)i such that lira
i
i~oo 3B(xj,rj)
(~,,,j)P dx
< 1.
(7)
Concentration phenomena
1023
Since f~(fte,,j) < I~p and lim inf 1 infO.D > O, ~--)0 +
~
t)
the sequence (u~i,j)i is bounded in H~'E(D). Therefore, (f%,j)i, or a subsequence, converges in H~'2(~) weakly, in L"(£)) and almost everywhere in D to a function fij. Since for every P1 . . . . , rk such that ~ > rj y j = 1 . . . . . k one has liminfinf_lA.i(x): x e ~ i~ ~ (.ei
B(xj, ~)
=
= +~,
it results k
~j=0
vx ~~
O B(xj, rj). j=l
Moreover, by (7) we have JB(xj,O (fiJ)P dx < 1, which implies
ID~jl 2 dx
~p < E r}2~+2p-'p)/" s= 1
B(xs,r,)
and so ~lp <
E
< --
[DYzj[ 2 d x
r(s 2 n + 2 p - n p ) / p
s= 1
B(xs,rs)
lim inf ~ -$ r(2n+2p-np)/P i--*oo
j= 1
[Dfl,i,j]2 dx B(xs,rs)
_< lim inff~,(fi,, j) in contradiction with the fact that lim inff~i(~,/) _
V j = l, ..., k.
Assume, by contradiction, that there exists an infinitesimal sequence of positive numbers (ei)i such that Jim
r(2"+2p-np)/p
l ~ ~o
t"
[Dflti,J 12 d x < t i p .
(8)
J B(xj , rj)
Then one can prove that the sequence (ue~,j)i is bounded in /_/~,z(~) and there exists a subsequence converging in LP(~) and almost everywhere in ~ to a function ~j which is zero in ~ \ B ( x j , rj). It follows that
llp < r>2n+2P-nP)/PI
IDflj[2 dx < lim r>2n+2p-nP)/Pf B(Xj, rj)
in contradiction with (8).
i ~ oo
,D~le,,j]2dx B(Xj, rj)
1024
D. PASSASEO
Property (V) is a simple consequence of the fact that
fe(tle,(k + l)) ~ ]flU, 2(P- 2)/p/lp] for every e > 0 small enough. Notice that this means, in particular, that the function /~e,(k+l) converges, as e ~ 0 ÷, to a nonnegative function t~tk+l) which differs from zero at most in two of the spheres B(xj, ri), j = 1 . . . . . k. •
Remark 3.3. The preceding theorem describes the effect of the concentration phenomena in uniformly elliptic problems, which approximate degenerate elliptic problems, in a simplified situation. It is clear that simple modifications of the proof enable one to obtain an analogous result in the case that, instead of the spheres B(xj, rfl, the connected components of the degeneration set are more general subsets Ci, with convex hulls two by two disjoined. However, assumptions of this type, which have an important role in the proof of theorem 3.2, are not necessary: in fact an analogous multiplicity result (proved in a paper in preparation) can be stated also when the degeneration set has a very general shape. REFERENCES 1. AMBROSETTI A. & RABINOWITZ P., Dual variational methods in critical point theory and applications, J. funct. Analysis 14, 349-381 (1973). 2. RABINOWITZ P., Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. math. J. 23, 729-754 (1974). 3. FABES E., KENIG C. & SERAPIONI R., The local regularity of solutions of degenerate elliptic equations, Communs partial, diff. Eqns 7, 77-116 (1982). 4. MURTHY M. K. V. & STAMPACCHIA G., Boundary value problems for some degenerate elliptic operators, Annali. Mat. pura appl. 80, 1-122 (1968) 5. STREDULINSKY E. W., Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Math. Vol. 107. Springer, Berlin (1980). 6. POHOZAEV S. I., Eigenfunctions of the equation Au + 2f(u) = O, Soviet. Math. Dokl. 6, 1408-1411 (1965). 7. BREZIS H. & NIRENBERG L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communs pure appl. Math. 36, 437-477 (1983). 8. BREZIS H. & NIRENBERG L., A minimization problem with critical exponent and non zero data, in Symmetry in Nature, pp. 129-140. Scuola Norm. Sup., Pisa (1989). 9. BREZIS H., Elliptic equations with limiting Sobolez exponents--the impact of topology, in Proceedings 50th Anniv. Courant lnst., Communs pure appl. Math. 39, 17-39 (1986). 10. LIONS P. L., The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana 1, 145-201; 45-121 (1985). 11. STRUWE M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities Math. Z. 187, 511-517 (1984). 12. PASSASEO D., Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math. 65, 147-166 (1989). 13. PASSASEO D., Nonexistence results for elliptic problems with supercriticai nonlinearity in nontriviai domains, J. funct. Analysis (to appear). 14. PASSASEO D., Risultati di non esistenza per equazioni ellittiche con nonlinearit~ sopraeritica. Preprint, Dip. Mat., Pisa (1992). 15. PASSASEO D., Existence e molteplicit~t di soluzioni positive per equazioni ellittiche con nonlinearit/l sopracritica. Preprint No. 619, Dip. Mat., Pisa (1992). 16. PASSASEO D., Esistenza e molteplicit/l di soluzioni nodali per equazioni ellittiche con esponente sopracritico in aperti contrattili. Preprint, Dip. Mat., Pisa (1992). 17. BAHRI A. & CORON J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Communs pure appl. Math. 41, 253-294 (1988). 18. CORON J. M., Topologie et cas limite des injections de Sobolev, Cr. hebd. S~anc. Acad. Sci. Paris Ser. 1 299, 209-212 (1984). 19. DANCER E. N., A note on an equation with critical exponent, Bull. London math. Soc. 20, 600-602 (1988).
Concentration phenomena
1025
20. DING W., Positive solutions of Au(n + 2)/(n - 2) = 0 on contractible domains, J. Partial diff. Eqns 2(4), 83-88 (1989). 21. PASSASEO D., Problemi ellittici con esponente critico. Forma del dominio e molteplicith di soluzioni positive. Preprint No. 564, Dip. Mat. Pisa (1990) 22. PASSASEO D., Su alcune successioni di soluzioni positive di problemi ellittici con esponente critico, Rend. Mat. Acc. Lincei 3, 15-21 (1992). 23. BENCI V. & CERAMI G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Archs ration. Mech. Analysis. 114, 79-93 (1991). 24. BENCI V., CERAMI G. & PASSASEO D., On the number of positive solutions of some nonlinear elliptic problems, in Nonlinear Analysis. A Tribute in Honour o f G. Prodi (Edited by A. AMBROSETTI and A. MARINO), pp. 93-107. Quaderni Scuola Norm. Sup., Pisa (1991). 25. CERAMI G. & PASSASEO D., Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with rich topology, Nonlinear Analysis 18, 109-119 (1992). 26. CERAMI G. & PASSASEO D., Existence and multiplicity results for semilinear elliptic Drichlet problems in exterior domains. Preprint, Dip. Mat., Pisa. 664 (1992).