Gazzola F., Radulescu V. - A nonsmooth critical point theory approach to some nonlinear elliptic equations in R^n(12).pdf

7/25/2019 Gazzola F., Radulescu V. - A nonsmooth critical point theory approach to some nonlinear elliptic equations in R^n(

1/12

A nonsmooth critical point theory approach

to some nonlinear elliptic equations in IR

n

Filippo Gazzola

Dipartimento di Scienze T.A. - via Cavour 84, 15100 Alessandria Italy

Vicentiu Radulescu

Department of Mathematics, Univ. of Craiova - 1100 Craiova Romania

Abstract

We determine nontrivial solutions of some semilinear and quasilinear elliptic problems on IRn;we make use of two different nonsmooth critical point theories which allow to treat two kinds ofnonlinear problems. A comparison between the possible applications of the two theories is alsomade.

1 Introduction

Consider a functional Jdefined on some Banach space B and having a mountain-pass geometry: the

celebrated theorem by Ambrosetti-Rabinowitz [1] states that ifJ C1(B) and Jsatisfies the Palais-

Smale condition (PS condition in the sequel) then J admits a nontrivial critical point. In this paperwe drop these two assumptions: in order to determine nontrivial solutions of some nonlinear elliptic

equations in IRn (n 3), we use the mountain-pass principle for a class of nonsmooth functionals which

do not satisfy the PS condition. More precisely, we consider a model elliptic problem first studied by

Rabinowitz [14] with the C1-theory and we extend his results by means of the nonsmooth critical point

theories of Clarke [6, 7] and Degiovanni et al. [9, 10]: one of the purposes of this paper is to emphasize

some differences between these two theories. This study was inspired by previous work on the existence

of standing wave solutions of nonlinear Schrodinger equations: after making a standing wave ansatz,

Rabinowitz reduces the problem to that of studying the semilinear elliptic equation

u+b(x)u= f(x, u) in IRn

(1)

under suitable conditions on b and assuming that fis smooth, superlinear and subcritical.

To explain our results we introduce some functional spaces. We denote by Lp the space of measurable

functionsu ofp-th power absolutely summable on IRn, that is, satisfyingupp:=IRn|u|

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b in (1) is greater than some positive constant; then we define the Hilbert space E of all functions

u : IRn IR with u2E :=IRn(|Du|

2 +b(x)u2) < . We denote by E the dual space ofE: as E is

continuously embedded in H1 we also have H1 E.

We first consider the case where () in (1) is replaced by a quasilinear elliptic operator: we seek

positive weak solutions u Eof the problem

n

i,j=1

Dj(aij(x, u)Diu) +1

2

ni,j=1

aij

s (x, u)DiuDju+b(x)u= f(x, u) in IR

n . (2)

Note that if aij(x, s) ij, then (2) reduces to (1). Here and in the sequel, by positive solution we

mean a nonnegative nontrivial solution. To determine weak solutions of (2) we look for critical points

of the functional J :E IR defined by

J(u) = 1

2

IRn

ni,j=1

aij(x, u)DiuDju + 1

2

IRn

b(x)u2 IRn

F(x, u) u E ,

where F(x, s) =s0 f(x, t)dt. Under reasonable assumptions on aij, b , f , the functional J is continuous

but not even locally Lipschitz, see [4]: therefore, we cannot work in the classical framework of critical

point theory. Nevertheless, the Gateaux-derivative of J exists in the smooth directions, i.e. for all

u Eand Cc we can define

J(u)[] =

IRn

ni,j=1

aij(x, u)DiuDj+

1

2

aij

s (x, u)DiuDju

+b(x)u f(x, u)

.

According to the nonsmooth critical point theory developed in [9, 10] we know that critical points u of

J satisfy J(u)[] = 0 for all Cc and hence solve (2) in distributional sense; moreover, since

n

i,j=1

Dj(aij(x, u)Diu) +b(x)u f(x, u) E

we also have1

2

ni,j=1

aij

s (x, u)DiuDju E

and (2) is solved in the weak sense ( E). We refer to [4] for the adaptation of this theory to

quasilinear equations of the kind of (2) and to [8, 11] for applications in the case of unbounded domains

and for further references. Under suitable assumptions on aij, b , f and by using the above mentioned

tools we will prove that (2) admits a positive weak solution.

Next, we take into account the case where f is not continuous: let f(x, ) Lloc(IR) and denote

f(x, s) = lim0

essinf {f(x, t); |t s|< } f(x, s) = lim0

esssup{f(x, t); |t s|< } ;

our aim is to determine u E such that

u+b(x)u [f(x, u), f(x, u)] in IRn . (3)

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Positive solutions u of (3) satisfy

0 I(u) ,

where

I(u) = 1

2

IRn

(|Du|2 +b(x)u2) IRn

F(x, u+) u E

and I(u) stands for the Clarke gradient [6, 7] of the locally Lipschitz energy functional I; more

precisely,

I(u) =

E; I0(u; v) , v , v E

,

where

I0(u; v) = lim supwu0

I(w+v) I(w)

. (4)

This problem may be reformulated, equivalently, in terms of hemivariational inequalities as follows:

find u E such that

IRn(DuDv+b(x)uv) +

IRn(F)

0(x, u; v) 0 v E , (5)

where (F)0(x, u; v) denotes the Clarke directional derivative of (F) atu(x) with respect tov(x) and

is defined as in (4). So, when f(x, ) is not continuous, Clarkes theory will enable us to prove that (3)

admits a positive solution.

The two existence results stated in next section have several points in common: in both cases we first

prove that the corresponding functional has a mountain-pass geometry and that a PS sequence can be

built at a suitable infmax level. Then we prove that the PS sequence is bounded and that its weak limit

is a solution of the problem considered; the final step is to prove that this solution is not the trivial

one: to this end we use the concentration-compactness principle [12] and the behaviour of the function

bat infinity. However, the construction of a PS sequence and the proof that its weak limit is a solution

are definitely different: they highlight the different tools existing in the two theories.

2 Main results

Let us first state our results concerning (2). We require the coefficients aij (i, j = 1,...,n) to satisfy

aij aji

aij(x, ) C1(IR) for a.e. x IRn

aij(x, s), aijs

(x, s) L(IRn IR) ;

(6)

moreover, on the matrices [aij(x, s)] and [saijs (x, s)] we make the following assumptions:

>0 such thatn

i,j=1

aij(x, s)ij ||2 for a.e. x IRn, s IR, IRn (7)

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(2, 2) , (0, 2) such that

0 sn

i,j=1

aij

s (x, s)ij

ni,j=1

aij(x, s)ij for a.e. x IRn, s IR, IRn .

(8)

We require that b Lloc(IRn) and that

b >0 such that b(x) b for a.e. x IRn

ess lim|x|

b(x) = + .(9)

Let be as in (8), assume that f(x, s)0 and

f : IRn IR IR is a Caratheodory function

f(x, 0) = 0 for a.e. x IRn

0 F(x, s) sf(x, s) s 0 and for a.e. x IRn

;

(10)

moreover, we require fto be subcritical

>0 f L2nn+2 (IRn) such that

|f(x, s)| f(x) +|s|n+2n2 s IR and for a.e. x IRn .

(11)

Finally, for all (2, 2) define q() = 2n2n+(2n) : then we assume1

C0 , (2, 2) , G Lq()(IRn) such that

F(x, s) G(x)|s| +C|s|2

s IR and for a.e. x IRn . (12)

In Section 3 we will prove

Theorem 1 Assume (6)-(12); then (2) admits a positive weak solutionu E.

Let us turn to the problem (3): we assume that f : IRn IR IR is a (nontrivial) measurable function

such that

|f(x, s)| C(|s| + |s|p) for a.e. (x, s) IRn IR , (13)

where C is a positive constant and 1 < p < n+2

n2

. Here we do not assume that f(x, ) is continuous:

nevertheless, if we define F(x, s) =s0 f(x, t)dt we observe that F is a Caratheodory function which is

locally Lipschitz with respect to the second variable. We also observe that the functional

(u) =

IRn

F(x, u)

1One could also consider the case = 2: in such case one also needs Gn/2 to be sufficiently small.

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is locally Lipschitz on E. Indeed, by (13), Holders inequality and the embeddingELp+1,

|(u) (v)| C(uE, vE)u vE ,

where C(uE, vE)> 0 depends only on max{uE, vE}.

We impose to f the following additional assumptions

lim0

esssup

f(x, s)s; (x, s) IRn (, )

= 0 (14)

and there exists >2 such that

0 F(x, s) sf(x, s) for a.e. (x, s) IRn [0, +) . (15)

In Section 4 we will prove

Theorem 2 Under hypotheses (9), (13)-(15), problem (3) has at least a positive solution inE.

Remark. The couple of assumptions (11) (12) is equivalent to the couple (13) (14) in the sense that

Theorems 1 and 2 hold under any one of these couples of assumptions.

It seems not possible to use the above mentioned nonsmooth critical point theories to obtain an existence

result for the quasilinear operator of (2) in the presence of a function fwhich is discontinuous with

respect to the second variable; indeed, to prove that critical points ofJ(in the sense of [9, 10]) solve (2)

in distributional sense, one needs, for all given Cc , the continuity of the mapu J(u)[], see [4].

Even ifJC1(E), we have at least JC1(W1,p E) forp 3nn+1 : this smoothness property in a finer

topology is in fact the basic (hidden) tool used in Theorem 1.5 in [4]; however, one cannot prove the

boundedness of the PS sequences in the W1,p norm. On the other hand, the theory developed in [6, 7]

only applies to Lipschitz continuous functionals and therefore it does not allow to manage quasilinear

operators as that in (2).

3 Proof of Theorem 1

Throughout this section we assume (6)-(12): by (6) and (8) we have

u E=n

i,j=1

aij

s (x, u)DiuDjuu L

1(IRn) (16)

and therefore J(u)[u] can be written in integral form.We first remark that positive solutions of (2) correspond to critical points of the functional J+ defined

by

J+(u) :=1

2

IRn

ni,j=1

aij(x, u)DiuDju+1

2

IRn

b(x)u2 IRn

F(x, u+) u E ,

where u+ denotes the positive part ofu, i.e. u+(x) = max(u(x), 0).

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Lemma 1 Letu E satisfyJ+(u)[] = 0 for all Cc ; then u is a weak positive solution of (2).

For the proof of this result we refer to [8]; without loss of generality we can therefore suppose that

f(x, s) = 0 s 0 , for a.e. x IRn

and, from now on, we make this assumption; for simplicity we denote J instead ofJ+.

Let us establish the following boundedness criterion which applies, in particular, to PS sequences2:

Lemma 2 Every sequence{um} Esatisfying

|J(um)| C1 and |J(um)[um]| C2umE

is bounded inE.

Proof. Consider {um} Esuch that |J(um)| C1, then by (10) we get

Im:=

1

2IRn

n

i,j=1

aij(x, um)DiumDjum

1

IRn f(x, um)um+

1

2IRn b(x)u

2

mC1 ;

by (16) we can evaluate J(um)[um] and by the assumptions we haveIRn

ni,j=1

aij(x, um)DiumDjum+1

2

IRn

ni,j=1

aij

s (x, um)DiumDjumum+

IRn

f(x, um)um+

IRn

b(x)u2m

C2umE .Therefore, by (8) and computing Im

1

J(um)[um] we get

2

2IRn

ni,j=1

aij(x, um)DiumDjum+ 2

2IRn b(x)u

2m C3umE+ C1 ;

by (7) this yields C4 >0 such that C4um2EC3umE+ C1 and the result follows.

Let us denote by Eloc the space of functions u satisfying(|Du|

2 + b(x)u2)< for all bounded open

set IRn and by Eloc its dual space; we establish that the weak limit of a PS sequence solves (2):

Lemma 3 Let{um} be a bounded sequence inE satisfying

IRn

ni,j=1

aij(x, um)DiumDj+1

2

IRn

ni,j=1

aij

s (x, um)DiumDjum= m, C

c

with {m} converging in Eloc to some Eloc. Then, up to a subsequence, {um} E converges in

Eloc to someu E satisfying

IRn

ni,j=1

aij(x, u)DiuDj+1

2

IRn

ni,j=1

aij

s (x, u)DiuDju= , C

c .

2We refer to [4, 9, 10] for the definition of PS sequences in our nonsmooth critical point framework.

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Proof. Asb is uniformly positive and locally bounded, for all bounded open set IRn we have

(|Du|2 +b(x)u2)<

(|Du|2 +u2)< ;

therefore the proof is essentially the same as Lemma 3 in [8]: the basic tool is Theorem 2.1 in [3] which

is used following the idea of [4].

The previous results allow to prove

Proposition 1 Assume that{um} E is a PS sequence for J; then there existsu E such that (up

to a subsequence)

(i) um u inE

(ii) um u inEloc

(iii) u 0 andu solves (2) in weak sense.

Proof. By Lemma 2, the sequence {um} is bounded and (i) follows. To obtain (ii) it suffices to apply

Lemma 3 with m= m+ f(x, um) b(x)umE wherem 0 in E

: indeed, ifum uin E, then

m in Eloc with = f(x, u) b(x)u. Finally, (iii) follows from Lemmas 1 and 3.

In order to build a PS sequence for the functional Jwe apply the mountain-pass Lemma [1] in the

nonsmooth version [10], see also Theorem 2.1 in [2]: let us check that Jhas such a geometrical structure.

First note that J(0) = 0; as the function F is superquadratic at +, we may choose a nonnegative

function e such that

e Cc , e 0 and J(te)< 0 t >1 .

Moreover, it is easy to check that there exist , > 0 such that < eE and J(u) ifuE = :

indeed by (12) we infer

IRn F(x, u) Gq()u

2+ Cu2

2 ;

hence, by (7) we have J(u) C1u2E C2u

E C3u

2Eand the existence of, follows.

So, Jhas a mountain pass geometry; if we define the class

:={C([0, 1]; E); (0) = 0, (1) =e} (17)

and the minimax value

:= inf

maxt[0,1]

J((t)) , (18)

the existence of a PS sequence for Jat level follows by the results of [9, 10].

We have so proved

Proposition 2 Let and be as in (17) (18); then J admits a PS sequence{um} at level.

As we are on an unbounded domain, the problem lacks compactness and we cannot infer that the above

PS sequence converges strongly; however, by using Proposition 1, the weak limit u of the PS sequence

is a nonnegative solution of (2): the main problem is that it could be u 0. To prove that this is not

the case we make use of the following technical result:

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Lemma 4 There existp (2, 2) andC >0 such thatu+mp C.

Proof. Using the relations J(um)[um] = o(1) and J(um) = +o(1), by assumptions (8) and (10) we

have

2 = 2J(um) J(um)[um] +o(1)

=

IRn

[f(x, u+m)um 2F(x, u+m)]

1

2

IRn

ni,j=1

aij

s (x, um)DiumDjumum+o(1)

IRn

f(x, u+m)um+o(1) .

Then, by (11), for all >0 there exists f L2nn+2 (IRn) such that

2IRn

|f(x)u+m(x)| +u

+m

22 :

as um2 is bounded, one can choose >0 so that

IR

n|f(x)u

+m(x)| . (19)

Now take r ( 2nn+2 , 2): then for all >0 there exist f L

r and f L2nn+2 such that

f= f+ f and f 2n

n+2 .

Then, by (19) and Holders inequality we infer

fru+mp+u

+m2

where p= rr1 ; as um2 is bounded, one can choose >0 so that

2 fru

+mp

and the result follows.

By the previous Lemma we deduce that {u+m} does not converge strongly to 0 in Lp. Taking into

account thatu+m2 andu+m2 are bounded, by Lemma I.1 p. 231 in [12], we infer that the sequence

{u+m} does not vanish in L2, i.e. there exists a sequence {ym} IR

n and C >0 such thatym+BR

|u+m|2 C (20)

for some R. We claim that the sequence {ym} is bounded: if not, up to a subsequence, it follows by

(9) that IRn

b(x)u2m +

which contradicts J(um) = + o(1). Therefore, by (20), there exists an open bounded set IRn

such that

|um|2 C >0 . (21)

So, consider the PS sequence found in Proposition 2; by Proposition 1, it converges in the L2loc topology

to some nonnegative function u which solves (2) in weak sense; finally, (21) entails u0.

The proof of Theorem 1 is complete.

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4 Proof of Theorem 2

In this section we assume (9) and (13)-(15); moreover, we set f(x, s) 0 for s 0.

To prove Theorem 2, it is sufficient to show that the functional Ihas a critical point u0 C, Cbeing

the cone of positive functions ofE. Indeed,

I(u) =u+b(x)u (u) in E ,

and, by Theorem 2.2 of [5] and Theorem 3 of [13], we have

(u) [f(x, u(x)), f(x, u(x))] for a.e. x IRn ,

in the sense that ifw (u) then

f(x, u(x)) w(x) f(x, u(x)) for a.e. x IRn . (22)

Thus, ifu0 is a critical point ofI, then there exists w (u0) such that

u0+b(x)u0=w in E .

The existence ofu0will be justified by a nonsmooth variant of the mountain-pass Lemma (see Theorem

1 of [15]), even if the PS condition is not fulfilled. More precisely, we verify the following geometric

hypotheses:

I(0) = 0 and v Esuch that I(v) 0 (23)

, >0 such that I on {u E; uE=} . (24)

Verification of (23). It is obvious that I(0) = 0. For the second assertion we need

Lemma 5 There exist two positive constantsC1 andC2 such that

f(x, s) C1s1 C2 for a.e. (x, s) IR

n [0, +) . (25)

Proof. From the definition we clearly have

f(x, s) f(x, s) a.e. in IRn [0, +) . (26)

Then, by (15),

0 F(x, s) sf(x, s) for a.e. (x, s) IRn [0, +) , (27)

where

F(x, s) =

s0

f(x, t)dt .

By (27), there exist R >0 and K1 >0 such that

F(x, s) K1s for a.e. (x, s) IRn [R, +) . (28)

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The inequality (25) follows now by (26), (27) and (28).

Verification of (23) continued. Choose v Cc (IRn) \ {0} so that v 0 in IRn; we obviously have

IRn

(|Dv|2 +b(x)v2)< + .

Then, by Lemma 5,

I(tv) = t2

2

IRn

(|Dv|2 +b(x)v2) (tv)

t2

2

IRn

(|Dv|2 +b(x)v2) +C2tIRn

v C1tIRn

v 0 large enough.

Verification of (24). First observe that (13) and (14) imply that, for any >0, there exists a constant

A such that

|f(x, s)| |s| +A|s|p

for a.e. (x, s) IRn

IR . (29)

By (29) and Sobolevs embedding Theorem we have, for any u E

(u)

2

IRn

u2 + A

p+ 1

IRn

|u|p+1 C3 u2E+ C4 u

p+1E ,

where is arbitrary and C4= C4(). Thus, by (9)

I(u) =1

2

IRn

(|Du|2 +b(x)u2) (u) C5 u2E C3 u

2E C4 u

p+1E >0 ,

for uE=, with , and sufficiently small positive constants.

Denote

={ C([0, 1], E); (0) = 0, (1)= 0 and I((1)) 0}

and

c= inf

maxt[0,1]

I((t)) .

Set

I(u) = minI(u)

E .

Then, by Theorem 1 of [15], there exists a sequence {um} Esuch that

I(um) c and I(um) 0 ; (30)

since I(|u|) I(u) for all u Ewe may assume that {um} C. So, there exists a sequence{wm}

(um) E such that

um+b(x)um wm0 in E . (31)

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Note that for all u C, by (15) we have

(u) 1

IRn

u(x)f(x, u(x)) .

Therefore, by (22), for every u Cand any w (u),

(u)

1

IRn u(x)w(x) .

Hence, if, denotes the duality pairing between E and E, we have

I(um) = 2

2

IRn

(|Dum|2 +b(x)u2m)

+1

um+bum wm, um +

1

wm, um (um)

2

2

IRn

(|Dum|2 +b(x)u2m) +

1

um+bum wm, um

2

2 um

2E o(1) umE .

This, together with (30), implies that the Palais-Smale sequence {um} is bounded in E: thus, itconverges weakly (up to a subsequence) in Eand strongly in L2loc to some u0 C. Taking into account

that wm (um) for all m, that um u0 in Eand that there exists w0 E such that wm w0

in E (up to a subsequence), we infer that w0 (u0): this follows from the fact that the map

uF(x, u) is compact from E into L1. Moreover, if we take Cc (IRn) and let := supp, then

by (31) we get

(Du0D+b(x)u0 w0) = 0 ;

as w0 (u0), by using (4) p.104 in [5] and by definition of ( F)0, this implies

(Du0D+b(x)u0) +

(F)0(x, u0; ) 0 .

By density, this hemivariational inequality holds for all Eand (5) follows; this means thatu0solves

problem (3).

It remains to prove that u0 0. If wm is as in (31), then by (22) (recall that um C) and (30) (for

large m) we get

c

2I(um)

1

2 um+bum wm, um=

1

2 wm, um

IRn

F(x, um)1

2

IRn

umf(x, um) . (32)

Now, taking into account its definition, one deduces that fverifies (29), too. So, by (32), we obtain

c

2

1

2

IRn

(|um|2 +A|um|

p+1) =

2 um

22+

A

2 um

p+1p+1 ;

hence,{um}does not converge strongly to 0 in Lp+1

. Frow now on, with the same arguments as in theproof of Theorem 1 (see after Lemma 4), we deduce that u00, which ends our proof.

Acknowledgements. This work was done while V.R. visited the Universita Cattolica di Brescia with a

CNR-GNAFA grant. He would like to thank Prof. Marco Degiovanni for many stimulating discussions,

as well as for introducing him to the critical point theory for continuous functionals.

F.G. was partially supported by CNR-GNAFA.

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Gazzola F., Radulescu V. - A nonsmooth critical point theory approach to some nonlinear elliptic equations in R^n(12).pdf