The equation −Δu+|u|α−1u=f, for0 ⩽ α ⩽ 1

0362~%X87 Pergamon

THE EQUATION

-Au

S3.00 + .@I Journals Ltd.

+ Iula% = f, FOR 0 s a < 1

THIERRY GALLOUET* Laboratoire d’Analyse Numerique, Tour Z-65, Stme Etage, Universite P. et IV. Curie, 75230 Paris Cedex 05, France

and CEREMADE,

Universitt Paris IX-Dauphine,

JEAN-MICHEL MOREL Place du Mar&ha1 de Lattre de Tassigny, 75775 Paris Cedex 16, France

(Receioed for publication 28 .May 1986)

1. INTRODUCTION LET r~ > 0 AND f~

L:,,(W),

n 2

1. We consider the following problem:

-Au + Iuln-’ u =f /@-i

in9’(W”), u E L!,,,(R”), (1)

U E L,‘,,(R”).

1

In [l], Brezis showed the following result. [l]. Let 1 < (Y< x. For every fE L,‘,,(W”), there exists a unique satisfying (1). Moreover, if fs 0 a.e., then K z 0 a.e.

u E LpOc(R”)

THEOREM

In this theorem, no limitation on the growth at existence of a solution u, and u is unique without proof uses a device introduced by Baras and Pierre solutions of (1). The preceding result fails if 0 < CYG 1. Indeed, Cauchy problem: u” =

ua

U’(0) = 0,

infinity of the data f is required for the prescribing its behaviour at infinity. The [2] which provides a local estimate of the it is enough to consider,

for n = 1, the

>

in R U(0) = ug.

(2) 1

One sees that for any u. > 0 (2) has a global positive solution in R, so that no uniqueness is expected even for positive solutions of (1). Let 0 < a < 1. We now briefly explain our plan and results: In Section 2 (theorem l), we prove that if f> 0 and (1) has a positive solution, then (1) has a “minimal” positive solution u (i.e. 0 G u c u for any other positive solution u). In Section 3, we focus on the problem of existence for 0 < (YC 1 of positive solutions of (1) for fE L:,,(R”), fa 0. This problem is closely related to the following (porous medium equation): form > 1, find U(X, t) 3 0 such that (au/at) = A(u”), * Present address: Dept. de Mathematiques,

u(x, 0) = uu(x).

(3)

Universite de Chambery, BP 1104, 73011 Chambery Cedex, France. 893

894

T. GALLOCJ%T~~~J.-M.MOREL

Indeed, the “implicit Euler scheme” associated with (3) is: u(t + h) - I

- hA~“(t + h) = 0,

u(r + h) 5 0, U(t) 2 0, h > 0.

(4)

Setting u(t + h) = h”‘“-’ tin, where (Y= m-l, we get: -Ali + d” =

h”‘l-nK(f),

u(r)

5

0,

ti

2

0.

(5)

We obtain precisely equation (1) with f = @‘-“u(r). Our main result is the following (theorem 2, Section 3): We set BR = {x E [w”,1x1< R}. For any 0 < CY< 1, there exists cr > 0 and c2 > 0 depending only on n and LYsuch that for fE L,‘,(R”), fa 0, one has: lim

sup R-n-(2~/1-~)

R-+=

1

fdx < ci) + (( 1) has a positive solution)

;

BR

((1) has a positive solution) +

lim”+“mp R-n-(2a/1-a)

(7)

Let us compare this result with the results obtained for the porous medium equation. It is known (cf. Benilan, Crandall and Pierre [3]) that if u,E L,b,(R”), u. 2 0, one has lam “+“pR-(2/m-1)-n --+ E

u. dx < m + ((3) has a solution on some interval [0, 7’1). >

(8)

The converse of (8) is proved in [4], (cf. also Dahlberg and Kenig [5]). It is straightforward to see that our result proves that for u. E L,‘,(W”), u. 3 0, the porous medium equation (3) has a local solution if and only if the “implicit Euler scheme” (5) (with t = 0, (Y= m-i) h as a positive solution u(h) for h > 0 small enough. It may be asked if it is possible to have ci = c2 in (6) and (7). In fact, we can prove, for n = 1, that no inequality of the kind liy:tp g(R)-’ JeR h(x) u(x) G C may be expected to be a necessary and sufficient condition in order to have positive solutions of (1) for fE Li,, f> 0. Many of the results stated above can be generalized: we give in Section 4 a proof that Brezis’ result ([l]) holds with any odd function g(u), instead of ]#-r U, which verifies: g is convex on R+, increasing, and I= ([a(t)dr)-“*dx<

+m.

(This kind of condition already arises in Osserman [6], who studies the inequation Au af(u)). For the case (Y< 1, our result can probably be extended to functions q(u), instead of ju]“-r U, with more general conditions. It may be asked whether equation (1) with 0 < LYC 1 has a solution for any f E L ~,(rW”). In fact we can prove that it has always infinitely many solutions (of course without condition of positivity). The proof involves different tools and will be done elsewhere [13]. In Section 5 we return to the case fa 0, u B 0, and deal with a characterization in some weighted Banach space of the minimal positive solution of (1). We obtain that if this solution verifies u(x) ]xI-2/1-rr+ 0 as Ix]+= + 30(that is the case if f(x) ]xI-*~/~-~+ 0), then every other

The equation -Au + luln-‘u = f, for 0 s a s 1

a95

positive solution of (1) verifies lim sup r-” - Czrril- a) r-+x.

I

u > 0. Br

We prove also by an example that this characterization

is not available if f(x) /xI-‘~“-~+

0.

Noration. In what follows, (Ydenotes always a real number such that 0 < cr < 1. If R is an open subset of R”, we often write: Jo u dr for Jo u(x) dr, where du is the Lebesgue measure in R”. By (1x1< R} we mean: {XE R”, 1x1s R}. We usually set B,= {Ixl < R}, S, = {Ix! = f}, and Js u da = Js, u(a) da, where da is the usual superficial measure on S,. If Q is an open set, I&p(Q) (k E R4,p > 1) is the Sobolev space of all measurable functions in Q whose derivatives in the distribution sense D’u verify D’u E U(Q) for 111< k. 9(Q) denotes the set of all C-functions with compact support in R. W$J’(Q) is the closure of 53(Q) in WkJ’(Q), and Wfj(R) is the set of all functions u such that Q~UE WkJ’(!2) for any q in Q(Q). For the Sobolev inclusions, we refer to Adams [8], and for the regularity properties of the Laplacian, to Agmon, Douglis and Nirenberg [9]. Finally for f and u E L,,‘,, “-Au + u= =f in 53?‘(Q)” means that for any Q, E ‘3(Q) one has: - Jo uAq, dx + Jo u”g, dr = Jo& du. This makes sense, since u E Lf,, * u” E Lb, for (Y< 1. The support of a function f will be denoted by suPP 5 2. PRELIMINARY

RESULTS

Let 0 < (Y< 1, f E L f,, (KY) with f 3 0. Consider the problem: Find u E L$,(W”), u 2 0, such that -Au + un =f in 53’(W”). (P) We shall say that u is a supersolution of (P) if u E L,‘,,([w”), u 5 0, and for some g E J%K(~“) -Au + ua = g >f

in

G3’(lRn).

We now prove the following result: THEOREM

1. Let 0 < LY< l,fE L:,,(K!“) withfa 0. If (P) has a supersolution u, it has also a “minimal” solution u < 0. (That is, a solution u such that 0 6 u s w for any other solution w of (P).)

In fact, theorem 1 is a consequence of the maximum principle. But the lack of regularity generates some difficulties, which we solve by using the following lemma (proved in the Appendix): LEMMA 1. Let u E W1.‘(BR), -Au E L’(B,), P = 0 on [w- and P’ L 0, then

-

AuP

and u G 0 on aBR. If P E C1 fl L”(R),

with

dx 2 0.

withgE Proof of theorem 1. Let u E L:,,,(Iw”) be such that u 20 and -Au+#=gaf> L~m(Iw”). We shall prove the existence of a solution u of (P) with u < u. For this purpose,

896

T. GALLOUETandJ.-M. MOREL

define the sequence (Qk of solutions of: -Auk + ]uk]=-r uk =f

(Pk)

in

9’(B,),

uk E w,j”(&).

The existence, uniqueness of uk and the fact that uk 5 0 (since fa 0) are proved in Brezis and Strauss [lo]. (Another proof can be found in [ll]). Note that uk - u verifies the assumptions of lemma 1 (with B = BR = Bk). Let P be as in lemma 1. Multiplying by P(uk - u) the inequality --A(&

-

U) +

IUkja-’

Uk -

IlIla-’

0 s

0

and then, integrating yields: iB

- A(u, - u) P(uk - u) dr +

IB

(]Ukla - I@) P(uk - 0) dr 6 0.

Thus by lemma 1: IB

(]&la - juj”) P(uk - u) ‘k 6 0,

and, choosing P> 0 on Iwy, this gives P(& - u) < 0 a.e., and then uk C u a.e. on B. By the same argument, one obtains uk G &+r on Bk. We conclude that (U& (and, therefore, (@)k) converges in L$,(R”). Passing to the limit in the distribution sense in (Pk) yields that u is solution of (P). Moreover 0 C II G u, and we obtain u G w for any solution w of (P), since w is also a supersolution.

3.THEMAINTHEOREM THEOREM 2. Let 0 C cy < 1. There exists ct > 0 and cZ > 0 (depending only on n and (u) such that for f E L{,,(Iw”), f Z 0, one has: (1) if liy2tp R -n-wl-@) la, f & < q, then (P) has a solution (and therefore a minimal

solution in the sense of theorem 1); (2) if (P) has a solution then li? “+“pR-“-(2”‘1-Lu .,-BRf dx < c2. -

I

3.1. Proof of part (1) of theorem 2 We first state the following lemma (“compact support lemma”) which is an essential tool in our proof. Roughly speaking this lemma is contained in [12]. For a sake of completeness we give a proof of this lemma in the appendix (cf. lemma A.3). We set for a, b E [w, Q(a, b) = {a S 1x1C b}.

COMPACT SUPPORT LEMMA (Appendix-lemma A.3). Let 0 < (Y< 1. For all 0 G a < b and for all E > 0 there exists C(E, a, b) > 0 such that if f E L{,(W),

f 3 0, Supp f C Q(a, b), 1 f dr s C(E, a, b),

(9)

The equation -Au + lulom’u=f, for 0 S (YS 1

897

then the problem -Au + ua =f

(P’>

in

9’(W)

u E L’(W), u 2 0, supp u c s2(a - E, b + E)

has a (unique) solution. Remark 1. Let R > 0. The function u is a solution of (P’) if and only if fi( .) = Rzl1-a u(. /R) is a solution of (P’) with!(.) = R2dleof( ./R), B = aR, 6 = bR, E = CR, instead off, a, b, E.

(The proof of this remark is straightforward.) Remark 2. From the compact support lemma and remark 1, one can easily deduce that for any f E ,!,I( W) with compact support, there exists a (unique) u E L ‘(R”) with compact support

and such that -Au + u” =fin

9’(W)

(cf. theorem 6.1 of [12]).

Using this compact support lemma we are going to prove the existence of a supersolution for (P), which, by theorem 1, is enough to ensure the existence of a solution for (P). To this end, we set, for k E N, s2k = R(24 2k+‘),

s2; = Q(O,2k).

Let g E r;&(W), g > 0, and such that for some K > 0, l$n s,“p R-n-(20’1-a) _ * one has for some K’ 2 0, g & s KR”+@ii-a)

+ K’.

JB, g clr < K. Then

(10)

I BR Thus, for k. > 1 large enough, we get gdu~2K(2ko)“+(*~/l-rr),

(11)

I n;, g & I

s 2”+(*“/1-~)

K.

(2k)n+W/l-a),

for

k

2

k,,_

(12)

Qk

If we assume that K s c; = 2- R- (2n’1-a) Inf(c(l/2,1,2), c(1/2,0, l)), we deduce from (11) and (12) (by using the compact support lemma and remark 1) the existence of {uk, k Z k. - 1) such that uk a 0,

g’nk if -AVk

+

Vg

supp ok c &_I uk,,-1

kak,

=

if

&a;,

SuPP

1

uk E L’(R”),

c

n;,

u Rk u ak+l, u

Qk,.

(13)

k=ko-1 k 3 k.

898

T. GALLOUET

andJ.-M.

~MOREL

Thus one sees that the series

i=kg-1

l=ko-1

converge a.e. and in L&(W”). Furthermore, c 3 0, one has

as (aa + 6” + c”) s 3(a + 6 + c)~ for all a, 6,

(14) Setting

we then deduce (with (13) and (14)): -Aw

w>o

+ 3~~ ag

in

9’(wn)

wELI’,,(R”).

(15) !

With z = 3iin-i w, this gives -AZ + zn 2 3+-i z E L,‘,,(R”),

*g

z 3 0.

(16) I

Setting now g = 3”r-Wf, we deduce from (16) that if

then z is a supersolution

of (P). This proves part (1) of theorem 2.

3.2. Proof of part (2) of theorem 2 Let u be a solution of (P). We first give a formal proof, that is assuming that u is regular enough. We then shall justify the calculations. Integrating on B, = B(0, t) the relation -Au + ua = f, and then integrating by parts, we get: (17)

We set

and obtain:

(18)

Theequation-Au+/@-‘u=f,forOscusl

Integrating

899

between 1 and r 2 1 and then integrating the first term by parts yields rn-i w(r).

~~rdt~~,~Q~~‘d~~~~u”+(n-l)~~‘~n-’w(r)+w(l)-

(19)

Thus, we have to estimate w(r). Using again (18) and integrating between 1 and r 2 1 yields: w(r) s

Thus, by

?!L_ i1

i

ua + w(1). 5,

Holder inequality: w(r) s IS1ll-a

11’ +I’

t”-’

w(t)”

w(t)” dt + r/l

t”-’

dt + w(l),

0

where ISi1 = Jsl da. Then, for r 3 1: w(r) < jS1j1-a [rjr

w”(t) dt] + w(l),

1

that is, again by the Holder inequality: w(r) G &2-n

(i,r Ilj:

drj

LT + C2r,

where c, only depends on n and CY,and cz is independent Set z(r) = a(b + r)(2/1-0)+1, where

(21)

of r.

,(l+-L)=C,a”(2+&)-‘.

One sees easily that for b large enough one has: z’(r)

2 Clr2-a(z(r))a

+ C2r,

r 3

1

z’(l) 2 w(1). We conclude that z(r) 2 w(r), t’r 2 1. Returning to (19) we make the same calculations as for obtaining (21) and get:

and using w(r) s z(r) we deduce that I dt

i1

f s c,(b

+ r)n+1+(2n11-a) + c5,

i Br

where c4 only depends on cy and n and C’sis independent nondecreasing, one has:

of r. But the function

for r 3 2.

(22)

t--+J’B, f being (23)

T. GALLO-T andJ.-M.MOREL

900

We conclude from (22) and (23) that: lim

r-n-(2n/1-a)

sup

l-P

I

fsc,, BR

with cr depending only on IXand n. To justify the calculations above, it is enough to have u E C’ n JVr~‘(lR”).This is not the case, but let Qikbe a sequence of mollifiers. One has

A.(u*

-

Q)k) + Ua * Q)k =f

+ Qjk,

and I.4* Q)kE C-(EP), Ua *~QI--+-,a

in Lf,,

f*P,k-,f

in L f,,

u*qk-+u

in Wf;:, and

then: u * q/( ---, u

(24)

in L1 (S,) for any sphere S,.

The relations (17), (18) h ave now a sense with u + Q, instead of u; un * Q, instead of ua, off. Then (19) and (20) are justified by letting k+ + ~0and using (24).

f + ~1instead

4.A GENERALIZATION THEOREM

3. Letg:R

--, R be continuous

=

OF BREZIS'RESULT[l]

odd function, convex on R+ and verifying

g(s>s2 0,

(25)

QssR.

dt

< TV, where G(s) = G(t) ‘I2

Then for any

f E L,‘,,(W)(n3

l), there exists a unique function u solution of

-Au + g(u) =

Moreover,

if f

30

f

in $!J’(R”)

(27)

a.e., then u L 0 a.e.

As in Brezis [l], the preceding result can be generalized THEOREM

1

4. Let h : R” x iw+

R be a Caratheodory

h(x, s) sgns 3 Ig(s)l,

in the following way.

function such that

Q.sE R,

a.e. inx E R”,

P-8)

where g verifies the assumptions of theorem 3, and: ;;-‘c9[/2(x, s)l E Lam

for any tin R.

(29)

The equation -Au + Iuj”-‘u = f, for 0 =SCYs 1

901

Then for every fE L,L,(R”) there exists at least a solution of -Au+h(*,u)

=f

in9’(W”), (30)

u E L,L,,([w”), A(., u) E L/0,@“). (We omit here the proof of theorem 4, which essentially is the same as in [l].) Remark 3. The results of theorems 3 and 4 are optimal in the following sense: Let g : R’

b

Ri+

be increasing, with g(0) = 0, and assume that g does not verify (26), that is: I= (~OSg(t)dt)-l”ds=~.

(31)

Then one sees easily that for any u. > 0, the O.D.E. -cf+g(u)=O

u(0) = uo, u’(0) = 0

1

has a positive solution in C*(R, R). Thus we have infinitely many solutions,

and no focal

estimate on them.

4.1. A local estimate for proving theorem 3 LEMMA 2. Let g verify the assumptions of theorem u E L:,JBR,) such that -Au + g(u) = f

3 and 0 < R < R’. Let f E L’(B,,)

and

in S’(B,,) (32)

g(u) E GX(&?). Then: i,,

ldu>ls c

(1+1

If I)

(33)

BR’

where C only depends on g, R and R’. In the case g(u) = 1~1~~~u, (Y > 1, the proof in [l] is based on a tool of Baras and Pierre [2] using Holder’s inequality. The following lemma will provide a generalization of this tool to a class of convex functions. 3. Let g : W++ W’ be a continuous odd function, convex on Rf and verifying g(0) = 0 and J” (j-6 g(s) d.~)-‘/~ dt < =. Then for any C > 0 there exists a convex function q E C*([O, 11, R+) verifying q(O) = q’(O) = 0 and

LEMMA

ww +@(x>)s &t)

v,(x)+ 2

for any x E [0, l] and t E R+. (9, depends only on g and C.)

(34)

902

T. GALLO~~T and J.-M. MOREL

Proof of lemma 3. Set:

for y > 0

k(Y) = [g-l (?)I-’

and k(0) = 0.

(39

We seek for a solution of the Cauchy problem: V(O) = q’(O) = 0,

q”(x) = k(cp(x)), q(x)>0

forxE]O,cu[,

(36)

cu>O.

Such a nontrivial solution exists since (26) implies I, (lo’/+)&)-“‘dt<=.

(37)

Indeed, (37) allows to define a solution (q,, cu) of (36) by:

J-o9cx) (Ior k(s) ds) -l’*dt = v’%

forx E [0, CY], with

,=min(~jd(~~‘k(~)di)-1’2dt,

I). I

(For a proof of (26) + (37), see lemma 4 below.) By (35) one has k(y)

1 g(lP(y))

- Y = 4c and, since t + g(t)/t is nondecreasing

k(y) < 1 g(t) --4c Y

t

l/k(y)

’

we get: foranyt,y

such that tk(y) 2 1.

E R+

Thus,

NY) <&g(t)y+l

foranyt,y>O.

(38)

Setting y = q(x) and using (36) yields t@‘(x) =G-$g(t)q(x)+l Extending

forxE[O,cy].

q to [0, l] by: q(x) = &a)

+ (x - c)(?J’(a) +

(x - a)* @“((u) 2 9

we get:

t@‘(x) s -&g(t)q(x)

+ 1

forx E [0, 11.

(39)

-Au

The equation

Note that QI”is nondecreasing

+ /u/~-‘u = f, for 0 S (Ys 1

903

on [0, 11. Thus I

q’(x) =

I

q”(t) dt s q”(x)

for x E [0, 11,

0

and, therefore,

(34) is a consequence

Proof of lemma 2. Multiplying

of (39).

(32) by sgn u and using Kato’s inequality we get:

For any r+~E ‘SJ(B,,) with t/ 2 0, we then have:

By a density argument, (40) remains true as rj.~E Cz(B,,) and I$ 3 0. Applying now (40) with q 0 $J instead of r/~,where 91 is given by lemma 2 and ij~= 1 on BR, 0 s q s 1 in BRs, We obtain:

Thus,

where C only depends on rj~. Using (34):

where Ci and C2 only depend on r@,R, R’, g. The use of (28) gives finally (33). (Note that rj~may be chosen in order to depend only on R and R’.) It remains to prove the relation (26) + (37): LEMMA

4. Let g

[g-‘(c/W’,

: R+*

w h ere r

c is

convex function and k : W +- 1W+defined by k(s) = some positive constant. Then

IR’ be a continuous

(~ofg(~)&)-l”df
implies10 (]~k(r)dr)e1~2

Proof of lemma 4. g and k being nondecreasing,

one has


T. GALLOUET~II~J.-M. MOREL

9c4

and the same inequalities for k. Thus (41) and ,-(~k(S)~)-1'2dr
~

~g-1(m11'2 i

0

cw@,-il,2k(cr)',2

Cc cNg'(u)

Cm@

$12

g(u)3'2
I

?I

@ [ - 2u1’*g(u)-qm

+

q30

where t = -

1

g(u)

du

u Wg( u) I/* < cD

(integrating by parts). Since g is convex, (g(t))/t is increasing, and then J” t-1’2g(t)-1/2 < ~0implies (g(t))/tas t+ + 30. Thus,

+ 30

This gives lemma 4. 4.2. Existence of solutions for theorem 3 The proof is now the same as in Brezis [l]. Instead of the Baras-Pierre lemma 1. So we just recall the main steps of the proof. (1) For any n E fW*there exists u, E W$‘(B,) solution of -Au, g(k)

+ g(u,) = f

estimate,

we use

in%‘(B,)

E NM.

1

(This is proved in [ 1 l] .) (2) BY lemma 2, (g(uJ>n is bounded in L&,(RN) and then Au,, and u, are bounded in &(RN) (indeed, k(t)] -> E1t 1f or some E > 0 and t large enough). Using the same method as in [l], we conclude that u,+ u and g(u,)+g(u) for some u in L~,,(lR”). Thus, u is solution of (30). This gives the part “existence” of theorem 3. 4.3. End of the proof of theorem 3 (1) LEMMA 5.Let g be as in theorem 3. There exists a family of functions (uR)RrRO such that: (i) uR E c’(B,), uR 2 0, uR(x) + + m as Ix]+ R; (ii) uR * 0 in L;“,(R”) as R-, + “0; (iii) -AU,

-I-g(uR)a 0.

Proof. Set H,,(u)

= J& (G(t) - G(uo))-‘I* dt, where G(f) = _l’bg(s) d.s, and uR(ug)(x)

= Hi,‘(a/xb,

The equation

-Au

+ /u/‘-‘u

=f,

for 0 C CYC 1

905

with R(u,) =

m (G(r) - G(u~))-“~

and u,, > 0.

(42)

UO

One sees easily that -AuRCuo) + g(uRCUo))2 0. Since g is convex, Jy G(t)-“’ dt = + so, and therefore R(u,) --, x as uO+ 0. Thus, we may define uR for any R 3 R0 with Ro large enough. The property (i) is clear, and to prove (ii), note that ~]xI

= /UR’X)(G(t) - G(u~))-~‘* dt

for R = R(u,).

(43)

UO

As R+ + J), one has by (42) u. + 0. As uo+ 0, (43) shows that uR(x) + 0 uniformly in x as x remains in a bounded set. (2) If u is a solution of (27), then f < 0 implies u s 0. Proof. Here again we follow the method of [l], using the comparison functions constructed in lemma 5.

One has by (iii): -A(u-uR)+g(u)-g(ua)
inBR.

Thus by Kato’s inequality -A(u

- uR)+ s 0.

(44)

Again by Kato’s inequality, we get -Au+ ==0 from -Auf + g(u)’ s 0. That implies u+ E L;“,,(R”). Then (i) yields (u - uR)+ = 0 near aBR, and this jointed to (44) gives u s uR a.e. By (ii) we obtain u s 0 a.e. (3) The solution u of (27) is unique. Let u and u be two solutions -A(u - u) + g(u) - g(u) = 0. Thus by Kato’s inequality:

of (27). Then

-A]u - u] + (g(u) - g(u)1 s 0. Note that if u and u have the same sign, ]g(u) - g(u)/ = 11’ Ig’(0ldta U

I’“-” Ig’(t)] dr = g(lu - u]), 0

and if uu < 0 and, for instance, ]u/ 3 ]uI:

I&w -&WI = Id4l + Id4 3 Ig(u)l2 lgpp)

1= /g(7)

/.

We then get: Ig(u) - g(u)1 L Ig((u - u)/2)], Vu, u E R. Thus, -A/u-u]+g By the preceding u = u a.e.

(I I) 7

~0.

result (2) applied with the function t+g(t/2),

we get ]u - uI c 0, that is

T. GALLOUPT and J.-M. MOREL

906

5. REMARKS ON THE BEHAVIOUR OF POSITIVE SOLUTIONS OF (1) AT INFINITY We saw in theorem 1 that if (1) has a positive solution forfz 0, it has also a minimal positive solution. An interesting question is as to whether this minimal solution can be distinguished from the others by its behaviour at infinity. (That is the case if (Y= 1.) We first give a positive answer to this question as the behaviour offat infinity is subcritical (i.e. f(.~)jxl-~“~(~-~)+ 0). THEOREM 5. Let u be the minimal positive solution of (1) for some f3 ~_x~-~~(*-~)~u(x)/ + 0

as

0, and assume that

Ix\--, + 00.

Then for every other positive solution u of (1) one has: lim sup r-+x:

r-n-(2/1-n)) i

u > 0. B,

Remarks. (1) If [~I-@~~(~-~)))f(x) + 0 at infinity, one sees easily that the supersolution constructed in the existence proof of theorem 2, and, therefore, the minimal solution u, verify Jx~-‘~(‘-~)u(x) --, 0 as 1x1--, + 00.Thus, as IxI-(~~‘(‘-~))f(x) + 0, we get a simple characterization of the minimal positive solution of (1). (2) If instead of Ixj-(2/(1-0))u(x) + 0 one only has ,-n-(2’(1-a)) Ia, u ---, 0, the result of theorem 4 seems to fail.

6. Let w E L:,,([w”) verifying IAwI 6 @. Then there exists a constant C depending only on (Ysuch that:

LEMMA

Vu

>

0,

(

sup r-(2/(1-n))-n r--t+=

lim

I

w

s

ln~~~

a I$

1x1-(Ql-@) w(x) s cu . 1

5,

Proof. Assume that lim sup r -(2/(1-o))-n lfl I u < u. Then for r 2 r. large enough one has r-5

r-w-4)-n i

u c a.

(45)

a,

Set now u,(x) = r- (2/(1-a)) u(rx). Then one sees immediately IAu,I G u;

and

i

u, G u.

c

cue

(46)

Bl

Using classical interior estimates and a straight bootstrap that: bri‘“(B~,~)

that u, verifies:

argument,

the relations (46) imply

with C depending only on (Y.

Thus, for 1x1G r/2, u(x) = r2’(1-a)u,(x/r) c Cu r2/(1-L1). We conclude that u(x) 6 C22/(‘-a)u~x~2~(‘-~). Proof of theorem 5. Let u be a minimal positive solution of (1) u(x) Ix1-(2/(1-o)) + 0 as Ix/+ + 00, and u + w another positive solution. Thus,

Aw = (u + w)” - Us 3 0.

such

that (47)

907

-Au + /u/*-'u =f, for0 G CYs 1

The equation

Assume by contradiction that r -(z;‘(l-a))-n Ja w--, 0 as r+ + 3c. Since (47) implies 1Awl < w”, we obtain byiemma 7 that 1x1-(z’(l-aJ)w(x) - 0 as IX-+ + x. If a, b 3 0, one verifies easily that there exists a constant C depending only on cy such that: b “).

(a + b)” 5 C inf(a”-‘6,

Applying this relation to u and w and taking into account w(x) 1x1-(“(l+) ---, 0 as IxI---, + 34, we obtain: v/E

>

IX/ > r,, =j (u +

ro,

0,

W)&

-

U=

2

that u(x)/xI-(‘,‘(~-~)) and (48)

CE-‘IX-‘W.

In what follows, we make formal calculations, assuming w as smooth as necessary. We shall justify them later. Integrating (47) on B, and using (48) yields: 2

i,,

I

if

I

(U + W)" -

U=

+

1x1-2

GE-l i

B’O

w.

(49)

BrBr,

Setting Co = Ja,, (U + w)~ - un > 0 if r. is large enough, we get

Set W(r) = -$

I S,

w. We obtain: r

rnel@‘(r)

L

CO(~)

+

CE-~

I

fne3W(t)

dt

for r 2 To(E).

r0

By Gronwall’s lemma, this implies that for any k > 0, r-%(r) --, + 00 as r--, + =, which contradicts the assumption IxI-(*‘(~-~))w(x) ---f 0 as IxI---, + 30. We now justify the above formal calculations: as in 3.2, we can use a sequence of mollifiers qk : we can write relations (47)-(48) with w * Q)~instead of w and ((u + w)” - u”) * qk instead of (u + w)” - P. With these modifications, (49) and (50) remain true. Call (5O)k the modified relation. Let now k+ + =. Since w E Wt,$ (R”), W E A.C. (W’, R), and passing to the limit in (5O)k yields (50) in the classical sense. The following proposition proves that if f has the limit behaviour at infinity which ensures existence of positive solutions, the result of theorem 4 is not available. PROPOSITION

1. There exists two positive constants a and b such that every positive solution

of -u”

+

ua = Cr241-a)

(51)

verifies a < lim sup Irl -(2/(1-a)) u(r) < 6. r-30

(52)

Proof. (1) By theorem 2, (51) admits positive solutions for c small enough. Thus the preceding proposition is nonempty. Assume now by contradiction that the left inequality in

T. GALLOU~Tand J.-M. MOREL

908

(52) is false. We get liy_szp 1r I- (VU-N) u(I) = 0, and by (1) : u”(r) = - c#(~-~) as r+ J). This implies u < 0 for r large enough and yields a contradiction. (2) From (51) we get U”s LP, and this implies easily the second inequality of (52). Acknowledgemenrs-H.

Brezis introduced us to this problem and suggested most of the results contained here. We thank also F. Mignot for helpful conversations. REFERENCES 1. BREZISH., Semilinear equations in (w”without condition at infinity Appl. Math. Optim. (to appear). 2. BARAS P. & PIERRE M., Singularites Climinables d’equations elliptiques semi-lineaires, Ann. Inst. Fourier (to appear). (Cf. also C.r. hebd. sPanc. Acnd. Sci. Paris 295, 519-522 (1982).) 3. BENILANPH., CRANDALLM. G. & PIERREM., Solutions of the porous medium equation under optimal conditions on initial values, Indiana Uniu. Math. I. (to appear). 4. ARONSOND. G. & CAFFARELLIL. A., The initial trace of a solution of the porous medium equation, Truns. Am. marh. Sot. (July 1983). 5. DAHLBERGB. E. T. & KENIG C. E., Nonnegative solutions of the porous medium equation, preprint. 6. OSSERMANR., On the inequality Au aflu), Pucif. 1. Marh. 7, 1641-1647 (1957). 7. DIAZ-DIAZ J. I., Solutions with compact support for some degenerate parabolic problems, Nonlinear Analysis 3,

831-847 (1979). 8. ADAMS R. A., Sobolev Spaces, Academic Press, New York (1975). 9. AGMON S., DOUGLISA. & NIRENBERGL., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Commune pure appl. Math. 12, 623-727 (1959). 10. BREZISH. & STRAUSSW. A., Semilinear second-order elliptic equations in L’, I. Math. Sot. Japnn 25, (1973). 11. GALLOU~~T T. & MORELJ. M., Resolution of a semilinear equation in Lt. Proc. R. Sot. Edinb. (to appear). (Cf. also C.r. hebd. s&nc. Acad., Sci. Parb Strie I, 296, 49W96 (1983).) 12. BENILANPH., BREZISH. & CRANDALLM. G., A semilinear equation in L’, Annuli Scu. norm. sup. Piss 2, 523555 (1975). 13. MOREL J. M., Runge property and multiplicity of solutions for elliptic equations in WNwith a monotone nonlinearity Manurcripta math., 54, 165-185 (1985).

APPENDIX In this appendix we prove several lemmas used in the preceding sections. Let n 3 1. LEE A.l. Let Q be a bounded regular open set of R” apd let u E wl.‘(S2) be such that u < 0 a.e. on aQ (in the sense of traces) and Au E L’(Q). Then there exists ut E a,(Q) such that II~4OonaQ,u,+uinL’(Q)andAu,+Au in L’(n). Proofoflemma A.l. Let fk E C-(n) and g, E C-(JQ) such that fk--f=

-Au

g,+g=u,,n There exists uq E c”(c)

inL’(R),

ask++=,

inL’(JR),

ask-*+m.

such that -Auk

=fk

Us =g,

onfi, on R.

I

Step 1. Estimates on uk.

Let q E

W.*(Q),

q

=

0 on &2 (1 < 4 < + 9). One has 1 f&x=-j P

Au&.x=-1 a

u,Ar+dx+//~$do. n

The equation -4u

909

+ [I((“-‘U = f, for 0 G a C 1

Then (Al) Let l/r E L”(R), there exists v E W*.q(Q) (1 s 4 < + 30) such that -4rp=y,

on R

cp=o and furthermore

(‘42)

0nac-2, I

one has (cf. [9]): IIVllW+ s CM,

s

CIIVII~

(where C denote some “constants” only depending on Q and 9). Taking some 9 > n we conclude (using the Sobolev injection theorem)

Then (Al) gives, for all v E L”(Q) G

c(l,fkfII

+

Ilgkl,l)ildl-~

Thus,

From this inequality (with uk - Us, instead of t+J we conclude that {ut, k E N} is a Cauchy sequence in L’(R). Then there exists v such that uk-, v in f.‘(Q). Slep 2. We show rhat v = u a.e. Let vk E B(Q) such that vk+ v in W’.‘(Q) and 4.~~+ 4v in L’(R). For cpE C’(6) such that QI= 0 on aS2, one has

Since -Avk+fin

L’(Q) and ok-g

iR

497v,dx=

-

I

acp

4VkV dx.

-Vkdr-

i

do an

0

in L’(dQ) ( smce uk-’ u in W’,‘(S2)) we conclude by letting k--+ + = that

- j

(A3) *

In (Al), let k+

+ p. Since uk+ v in L](Q) we also have

I

vA~dx=

(A4)

n

(A3) and (A4) give: Io

(u-v)Ap,&=O

(W

for all Q, E C?(n) with q~= 0 on aS2. Let v E B(Q). There exists Q, E C(6) such that 497 =* q =O

on

R

on

a52

T$s (A5) gives Jo (u - v) w d_x= 0 for all y E PI(R), and, therefore,

u = v a.e.

T. GALLOUETand J.-M. MOREL

910

LEMW A.2. Let Q be a bounded regular open set of R”. Let u E W’,‘(Q) -Au E L’(Q). Let P E C’ fl L”(R), P = 0 on iw-, P’ 2 0. Then one has

IR Proof of lemma

be such that u s 0 a.e. on aQ and

-Au P(u) dx B 0.

A.2. By lemma A. 1 there exists (ut, k E N} C 9(G) such that lli -+ U

in

-Au,, + -Au in

L’(Q)

as

k-t+=,

.L.‘(Q)

as

k-+x,

ac2.

on

UJ=so

We clearly have IQ

- Au, P(u~) =

We can assume (after extracting a subsequence) ilk +u

IAuk]
dx 2 0.

that a.e.

-Au,, + -Au

By the dominated convergence

I~u,[* P’(Q)

a.e.

a.e.,

with some

g E L’(Q).

theorem we conclude that - Au P(u) dr z= 0.

For a, b E R, we set S2(a, b) = {a S 1x1< b}. LEMMA A.3. (Compact support lemma-cf. [12, theorem 6.11 and [7].) Let 0 < (Y< 1. For all 0 < (I < b and for all E > 0, there exists C(E, a, b) > 0 such that if

f~ Lk(IW”),f~

0, SuppfC

Q(a, b), I/b

c C(E, a, b),

(9)

then the problem -Au+u”=f

in

a’;w”)

u E L’(W”), u aO,SuppuCR(a-e,b+&)

(P’) I

has a (unique) solution.

of

Proof lemma A.3. Let 0 < (Y< 1. We prove lemma A.3 with, for instance, a = 2, b = 3, and some 0 < E < 1. (This case contains all the difficulties.) We set 52 = Q(2,3), B = {l < 1x1< 4) and QL = R(2 - E, 3 + E). Let fE L’(R”), f* 0, SuppfC R and u be the unique solution (cf. [lo]) of -Au + u” = f UE W$‘(B),

?iY(B) uz=o. I

(p”)

Our aim is to prove that for JafrLx small enough one has Supp u C R,. We then obtain of course a (unique) solution for (P’). Step 1. Estimates on u. One has (cf. [IO]) la u”cLr < .fafcLr and then Is ]Au] dx < 2 fef&, (only depending on n)

this gives for some C,

(‘46)

911

The equation -AU + 1~1‘~‘~= f, for 0 c (Y<

Since Au = u” on Ati, a classical “bootstrap” argument gives u E Wz(Afi) (and then, takingp > n. u E C’(A\ii)). Let 1 < R, < RI < 2 < 3 < R3 < R, < 4. Then there exists CL (only depending on n, R,, R2, R,, R,) such that or

Sup{u(x), R, s 1x1s R2

R, s 1x1s R,} s C2

JB

fb.

In what follows we take R, = 1 + ~14, R, = 2 - ~14, R, = 3 + E/-I, R, = 4 - ~14. Sfep 2. Estimnres on Supp u. In this step we construct on {I < 1x1G RJ and {R, =s 1x1< 4} some radial comparison functions. Let M, = Max{u(x), 1x1= RJ. One has, by step 1,

We set, for some A > 0 and R, = RI - ~14 = 2 - ~12,

u(x) = A(Ixl - R,)z’(‘-“)

if

RSSjxj~R,

u(x) = 0

if

1 < 1x1< RS,

an easy calculation yields -Au(x)

+ u”(x) = A” -A

2(1 + LY) w)

(1x1- Rs)‘=‘(‘-“)

-?-(G,

-&n-l)

_ Rs)(l+d(l-“)

=0

if

1 < 1x1=GRj.

Since R, - R5 < 1 and 0 < (Y< 1 one has

i

(l-4- Rs)U+u)/(l-0)

<

_!-(lxl - Rj)W-@

if

RS s 1x1s R,,

that is

Taking A = A, > 0 small enough (A, only depends on n, cy and E) one has -Au+u”~O

on

l+l~Rz.

(A7)

Taking also fdx+Ac(R2 2

- R5)2’(‘-a)

= C:’ >O,

(A8)

one has if

1x1= R2.

649)

(A7) and (A9) then give: -A(u - u) + u” - u” =G0

on

1 < Ix/ G R,

u - u E W’,r(l < 1x1CR,) u - u

Thus,

s 0 on

(1x1= 1) U {lx1= R,}

I

by lemma A.2, one has for P E C’ II L”(R), P = 0 on Iw-, P > 0 on rW,*, P’ P 0, (u” - 00) P(u - u) =s 0.

This proves u(x) 6 u(x) if 1 < 1x1CR,.

912

T. GALLO~T and J.-M. MOREL

Then we conclude u(x) = 0 if 1 < Ix/ G R, = 2 - E/Z. By a similar method (a little bit simpler) we prove that there exists CF* that if

> 0 (only depending on n, (Yand E) such

.lO)

then U(X) = 0

if

3 + ~/2 = R6 s 1x1< 4.

Hence, if f satisfies (A8) and (AlO) (that is if Jsfd.r 6 C, = Inf(C3, SuppucR,=R(2-E,~+E). This proves lemma A.3 for (I = 2 and b = 3.

Ct*))

one has