GENERAL QUASILINEAR PROBLEMS INVOLVING P(X)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 30-41

DOI: 10.15826/umj.2020.1.003

GENERAL QUASILINEAR PROBLEMS INVOLVING p(x)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

Hassan Belaouidel1, Anass Ourraoui2, Najib Tsouli3

Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco

Abstract: This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving p(x)-Laplace type equation, namely

i -div(a(|V«|p(x))|V«|p(x)-2V«) = Af(x,u) in Q,

\\ n ■ a(|VM|p(x))|V«|p(x)-2VM + b(x)|u|p(x) 2u = g(x,u) on dQ.

Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.

In this paper we study the nonlinear elliptic boundary value problem with Robin conditions

J -div(a(|Vu|p(x))|Vu|p(x)-2Vu) = Af(x,u) in Q,

\\n • a(|Vu|p(x))|Vu|p(x)-2Vu + b(x)|u|p(x)-2u = g(x,u) on dQ, (L1)

where Q is an open bounded subset of RN (N > 2), with smooth boundary, n is the outer unit normal vector on dQ, b is a positive continuous function defined on Rw, p € C+(Q) with

and p(x) < p*(x) where

p*(x) = I N-p(x) i y &

if p(x) > N

for any x € Q,. It is clear that the equation in question is elliptic since it describes phenomena that do not change from moment to moment, and that the operator

Lu = -div (a(|Vu|p(x))|Vu|p(x)-2Vu)

is an elliptic operator in divergence form.

Recently, the study of differential equations and variational problems involving p(x)-growth conditions have been extensively investigated and received much attention because they can be presented as models for many physical phenomena which arouse in the study of elastic mechanics [32], electro-rheological fluid dynamics [27] and image processing [6], electrical resistivity and

polycrystal plasticity [3, 4] and continuum mechanics [2] etc, for an overview of this subject, and for more details we refer readers to [11] and [5, 10] and the references therein. The existence of nontrivial solutions to nonlinear elliptic boundary value problems has been extensively studied by many researchers [1, 7, 14, 15, 18, 21, 23, 24] and references therein.

It is known that the extension p(x)-Laplace operator possesses more complicated structure than the p-Laplacian. For example, it is inhomogeneous and usually it does not have the so-called first eigenvalue, since the infimum of its spectrum is zero.

However, to understand the role of the variable exponent, well, although most of the materials can be accurately modeled with the help of the classical Lebesgue and Sobolev spaces Lp and W1,p, where p is a fixed constant, there are some nonhomogeneous materials, for which this is not adequate, e.g. the rheological fluids mentioned above, which are characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field. Thus it is necessary for the exponent p to be nonstandard, therefore, the spaces with variable exponents are required. As an introduction and a history coverage to the subject of variable exponent problems, we advice the reader to see the monograph [12] and the articles [16, 20].

Note that, the p(x)-Laplace operator in (1.1) is a special case of the divergence form operator —div (a(|Vu|p(x))|Vu|p(x)-2 Vu) which appears in many nonlinear diffusion problems, in particular in the mathematical modeling of non-Newtonian fluids. When

a(t) = 1 + t

vT+F

we have the generalized Capillary operator (which is essential in applied fields like industrial, biomedical and pharmaceutical) initiated by W. Ni and J. Serrin [22].

Inspired by the works in [25] and [19], we study the existence and multiplicity of nontrivial solutions the problem (1.1), via the mountain pass theorem and the Rabinowitz&s symmetric mountain pass theorem [26].

We assume the following conditions:

(Ao) The function a : R+ — R is continuous and the mapping © : RN — R, given by ©(C) = A(|{|p(x)) is strictly convex, where A is the primitive of a, that is

A(t) = [ a(s)ds. o

(Ai) There exist two constants 0 < L < K such that L < a(t) < K for all t > 0.

We assume that f,g : Q x R — R are of Caratheodory functions, f (x, ■) = g(x, ■) = 0 and satisfy:

(Fo) for all (x,t) € Q x R |f(x,t)| < fi(x)|t|r(x)-i, such that

where f\\ is nonnegative, measurable function and f\\ € Lp(-x)-t(-x) (Q); (Fi) for all (x,t) € Q x R |f(x,t)| > f2(x)|t|a(x)-1,

where f2 > 0 in some nonempty open set O C Q;

(Go) for all (x,t) € dQ x R, |g(x,t)| < gi(x)|t|q(x)-1,

(N - l)p(x) pd(x) = (p(x))d = { N-p(x)

+o if p(x) > N

and there exists a positive constants Cg such that 0 < g1 < Cg ;

(Gi) for ail (x,t) edQxR lim = o.

v 1y v y t^o |t|p+-1

(G2) there exists ^ > p+ such that ^G(x, t) < g(x, t)t for all (x, t) € dQ x R, where

G(x,t) = / g(x,s)ds. o

The main result of this paper is as follow.

Theorem 1. Assume that (A0)-(A1), (F0)-(F1) and (G0)-(G2) hold. Then there exists X* > 0 such that for every X €]0, X*[, the problem (1.1) admits at least one nontrivial .solution. In addition, if we assume the following conditions :

(G3) there is a nonempty open set U C dQ with G(x, t) > 0 for all (x, t) € U x R+, (G4) the functions f and g are even,

then the problem (1.1) has infinitely many solutions for every X > 0.

The remainder of this paper is organized as follows, in Section 2 we introduce some technical results and required hypotheses in order to solve our problem, in Section 3 we state some and prove the main results of this work.

In the sequel, let p(x) € C+(Q), where

C+(Q) = {h : h € C(Q), h(x) > 1 for any .ïéÔ}. The variable exponent Lebesgue space is defined by

Lp(x)(Q) = {u : Q ^ R measurable and J |u(x)|p(x) dx < +o}

furnished with the Luxemburg norm

Il II ri f u(x) P(x) 1

M£p<*)(n) = Mp(®) = inf |<7 > 0 : J& dx < 1 j.

Remark 1. Variable exponent Lebesgue spaces resemble to classical Lebesgue spaces in many respects, they are separable Banach spaces and the Holder inequality holds. The inclusions between Lebesgue spaces are also naturally generalized, that is, if 0 < mes (Q) < oo and p, q are

variable exponents such that p(x) < q(x) a. e. in Q, then there exists a continuous embedding Lq(x) (Q) ^ Lp(x) (Q).

The variable exponent Sobolev space is defined by

W 1>p(x)(Q) = {u € Lp(x)(Q) : |Vu| € Lp(x)(Q)}

equipped with the norm

llullw !.p(x) (Q) = |u|LP(x)(Q) + |Vu|LP(x)(Q}Proposition 1 [16, 17]. The spaces Lp(x)(Q) and W 1,p(x)(Q) are separable, uniformly convex, reflexive Banach spaces. The conjugate space of Lp(x) (Q) is Lq(x) (Q), where q(x) is the conjugate function of p(x), i.e.,

for all x € Q. For u € Lp(x)(Q) and v € Lq(x)(Q) we have

[ u(x)v(x)dx < (4r + 4:)Mp(s)M«(s)-,/n \\p q y

Moreover, if hi,h2,hs : —>■ (l,oo) are Lipschit.z continuous functions such that

/?i h2 h3

then for any u € Lhl(x)(Q), v € Lh2(x)(Q), w € the following inequality holds (see [ ,

Proposition 2.5])

/ < (— + — + — j|t/.|/ll(.T)|v|/l,(.T)|w|/l3(.T)Jn vft1 h2 h3 /

Proposition 2 [13]. Let p(x) and q(x) be measurable functions .such that p(x) € Lœ(Q) and 1 < p(x)q(x) < for a.e. x € Q. Let u € Lq(x)(Q), u = 0. Then

|u|p(x)q(x) < 1 ^ |u|p(x)q(x) < IM^ |q(x) < |u|p(x)q(x), |u|p(x)q(x) > 1 ^ |u|p(x)q(x) < U*) < |u|p(x)q(x)"

In particular if p(x) = p is a constant, then

||u|P|q(x) = |u|pq(x)Proposition 3 [16, 17]. Assume that the boundary of Q possesses the cone property and p,r € C+(Q) such that r(x) < p*(x) (r(x) < p*(x)) for all x € H. then there is a continuous (compact) embedding

W 1&p(x)(Q) ^ Lr(x)(Q),

Proposition 4 [9]. For p € C+(Q) and such r € C+(<9Q) f/iaf r(x) < pd(x) (r(x) < pd(x)) for all x € H. i/iere is a continuous (compact) embedding

W 1&p(x)(Q) ^ Lr(x)(dQ).

Proposition 5. [8, Theorem 2.1] For any u € W 1,p(x)(Q), let

||u||d := |u|LP(x)(SQ) + |Vu|LP(x)(n).

Then ||u||d is a norm on which is equivalent to

lulff l>p(x) (Q) = |u|LP(x)(Q) + |Vu|Lp(x)(Q).

Now, for any u € X := W 1&p(x)(Q) define

Vu(x) p(x) f u(x)

Il II ■ fi n I Vu(x) p(x) / u(x) p(x) ,

||«|| := mf i a > 0 : / -— dx + / b(x) dax < 1 k

L Jn V Jdn V J

where b € Lœ(Q) and dvx is the measure on the boundary dQ. Then by Proposition 5, || ■ || is also a norm on W 1,p(x)(Q) which is equivalent to || ■ ||Wi,P(x)(n) and || ■ ||d, the proof of this statement can be found in [8, p. 551]. Now, we introduce the modular p : X ^ R defined by

p(u)= / |Vu|p(x)dx + / b(x)|u(x)|p(x) dvx Jn Jdn

for all u € X. Here, we give some relations between the norm || ■ || and the modular p. Proposition 6 [16]. For u € X we have

(i) ||u|| < 1(= 1; > 1) ^ p(u) < 1(= 1; > 1);

(ii) If ||u|| < 1 ^ ||u||p+ < p(u) < ||u||p- ;

(iii) If ||u|| > 1 ^ ||u|p- < p(u) < ||u||p+.

Proposition 7 [29]. Suppose that f : Q x R ^ R is a Carathéodory function and satisfies the growth condition

|f (x,t)| < c|t|a(x)/^(x) + h(x), for every x € Q, t € R,

where a, p € C+(Q), c > 0 is constant and h € L^x)(Q). Then Nf(La{-x)(Q)) ç Ll3^(Q), where Nf (u)(x) = f (x,u(x). Moreover, Nf is continuous from La(x)(Q) into L^(x)(Q) and maps bounded set into bounded set.

As a consequence of Proposition 7, the Carathéodory function f defines an operator Nf which is called the Nemytskii operator.

Definition 1. We say that u € X is weak solution of (1.1) if

I a(|Vu|p(x))|Vu|p(x)-2VuVvdx + / b(x)|u|p(x)-2uvdvx = X / f(x,u)vdx + / g(x,u)vdvx Jn Jdn Jn Jdn

for all v € X.

Now we introduce the Euler-Lagrange functional : X —> R associated with problem (1.1) defined by

h(u)= I -¡—A(\\Vu\\p{x))dx + I -j—b(x)\\u\\p{x) d,crT - A I F(x,u)dx - I G(x,u)daT, Jn P(x) Jdn P(x) Jn Jdn

F(x,t) := / f(x,s)ds. Jo

Furthermore, the (weak) solutions of (1.1) are precisely the critical points of the functional

Lemma 1 [31] . Let

L(u) := [ -j—A(\\Vu\\p{x))dx + [ -j—b(x)\\u\\p{x)daT. Jq P(x) JdQ p(x)

Then the mapping L : X ^ X* is a strictly monotone, continuous bounded homeomorphism and is of type (S+), namely assumptions un ^ u and limsup L(un)(un — u) < 0, imply un ^ u.

By Proposition 7, we can see that the functional is well defined on X and € C 1(X, R) with its Frechet derivative is giving by

fx(u) ■ v = / a(|Vu|p(x))|Vu|p(x)-2VuVvdx + / b(x)|u|p(x)-2uvda^

—A / f(x,u)vdx — / g(x,u)vd<rx ./q ./sq

for all u, v € X.

Let X be a real Banach space and let be a functional / € C 1(X, R). We say that / satisfies the Palais-Smale condition on X ((PS)-condition, for short) if any sequence (un) C X with (/(un)) bounded and /&(un) ^ 0 as n ^ to, possesses a convergent subsequence. By (PS)-sequence for / we understand a sequence (un) C X which satisfies the conditions: (/(un)) is bounded and /&(un) ^ 0 as n ^ to.

The main tools used in proving Theorem 1 are the well known mountain pass theorem and its the symmetric mountain pass theorem.

Theorem 2 [26, Theorem 2.2]. Let X be a real Banach space and let / belong to C 1(X, R) satisfying the (PS)-condition. Suppose that /(0) = 0 and that the following conditions hold:

(11) there exist p > 0 and g > 0 such that /(u) > g for ||u|| = p;

(12) there exists e € X with ||e|| > p such that /(e) < 0. Let

r = {7 € C([0,1]; X) : 7(0) = 0,7(1) = e}, c = inf mi« /W)),

then, c is a critical value of /.

Theorem 3 [28, Theorem 2.1]. Let X be a real Banach space and let / belong to C1(X,R) be even, satisfies (PS)-condition and /(0) =0. If X = Y ® Z with dim Y < to, and / satisfies

(I&1) there are constants p, > 0 such that //dBpnZ > 0

(I&2) there a finite dimensional subspace W C X, with dim Y < dim W < to and there is M > 0

such that max /(u) < M «ew

(I&3) considering M > 0 given by (I&2), / satisfies (PS)c for 0 < c < M.

Then / possesses at least dim W — dim Y pairs of nontrivial critical points.

To prove Theorem 1 we recall some lemmas presented below.

Lemma 2. Assume that (A1), (Fo) and (G2) hold. Then the functional satisfies the PalaisSmale condition on X ((PS)-condition, for short) at any level d.

Proof. Let d € R and let (un) C X be (PS) sequence for /A, then

/\\(un) ^ d and /&x(un) ^ 0 as n ^ to. (3.1)

First, we prove that sequence (un) is bounded in X. Suppose (un) unbounded, we may assume ||un|| ^ +to as n ^ to.

By (2), (A1), (F0) and Proposition 6 we have

h(un)= [ -±-A(\\Vun\\p(-x))dx+ [ -±-b(x)\\un\\>^dax Jq p(x) JdQ p(x)

—A / F(x,un)dx — / G(x,un)dax

jq jdQ (3 2)

>4/ [ -Lb(x)\\un\\p(-X)dax-^ [ fi(x)\\un\\r^dx- [ G(x,un)dax

p+ JQ JdQ p+ ./Q ./dQ

> - A f f1(x)\\un\\r{x)d,x - [ G(x, un)dax.

P+ r+ Jq JdQ

From (3.2), (Fo) and Proposition 6 we obtain

-l&x(un) • Un = - f a{\\Vun\\^x))\\Vun\\^x)dx + - [ b(x)\\un\\^x)dax № № J Q № J dQ

— / f{x,um)undx--/ g(x, un)und(jx (3.3)

№ J Q № J dQ

> ^ ||-ura||p -- [ fi(x)\\un\\r{x)dx - - [ g(x, un)und(jx.

№ № ./ Q № ./ dQ

Meanwhile, according to (F0), Proposition 4 and Proposition 2 it yields

I h(x)\\un\\r^dx< f \\fi(x)\\\\un\\r{x)dx < |/i| P(I) |u„|

•/Q 7n Lp(x)-r(x) (Q1

< l/il P(*) max ( K/,ra|lT),\\un\\ptx) ) < |/i| p(x) max [Cr- ||t/,n||r , Cr+||u„||&r+)

Lp(x)-r(x) (Q) V V LP(i)-r(i)(fi) V /

where Cr- and Cr+ are constants of compact embedding X ^ Lp(x)(Q). Using ( ), ( ), ( ), (3.4) and (G2) we obtain

d + 1 + i|| > h(un) - -l&x(un).un > ^ ||ura||p - 4 [ fi(x)\\un\\r{x)d,x

min(L, 1) - A f {x) 1

I gy x, un ) u,nuax "" (3.5)

G(x,un)dax---—\\\\un\\\\p — / fi(x)\\un\\r{x)d,x— / g(x,un)undax

, .....- .. -ir— / fi(x)\\un\\nx

/dQ № № ./ Q № J dQ

> min(L, l)f4~~)llu"HP -f4+-) ( h(x)\\un\\r[x]dx+( (-g(x,un)un-G(x,un))dax > min(L, 1)( —--)\\\\un\\\\p — ( —j—I— ) I/11 p(x) Cr+\\\\un\\\\r+,

VJ9+ Jli/" Vr+ (Q)

where d is defined in (3.1). Since p- > r+ (un) is bounded.

Now, with standard arguments, we prove that any (PS)d sequence (un) in X has a convergent subsequence. Indeed, the space X is a Banach reflexive space then there exists u € X such that, up to subsequence still denoted by (un) and by the Sobolev embedding, we obtain:

• un ^ u in X as n ^ œ;

• un(x) ^ u(x) a.e. in Q as n ^ œ;

• un ^ u in as n ;

• un —^ t/> in Lp^&-^Q) as ??.—>• oo. □

Proposition 8. If un ^ u in X as n ^ œ, then

lim [ /i(x)|u„|r(x)-1(u„ - u)dx = 0, (3.6)

n^œ./ n

lim f gi(x)|u„|q(x)-1(u„ - u)dax = 0. (3.7)

n^œ ,/dn

Proof. To demonstrate (3.6), we use Propositions 1-4 we give

I /i(x)|u„r(x)-1(u„ - u)dx < / |/i(x)||u„|r(x)-1|un - u|dx nn

< 3C |/i| max (\\un\\rp{x) \\ \\un\\rp+{x) K ~ u\\ {x),

T p(x)-rix) iCl\\ V 1 v & J V & /

where C is positive constant. By the compact embedding X ^ and the inequality

||un|p(x) - |u|p(x)| < |u„ - u|p(x), we obtain |u„ - u|p(x) ^ 0 in Lp(x)(Q) and |un|p(x) ^ |u|p(x). Similar arguments establish (3.7). Now, in virtue of (3.1) and Proposition 8, we have

lim sup / a(|Vu„|p(x))|Vu„|p(x)-2Vu„(Vu„ - Vu)dx + / b(x)|u„|p(x)-2u„(u„ - u)da^ n^œ Jn ./dn

= lim sup !^(un) ■ un + lim sup A / /(x,un)(un - u)dx + lim sup / g(x,un)(un - u)dax = 0. n^œ n^œ Jn n^œ Jdn

Finally, by Lemma 1 un ^ u in X.

To finish the proof of the Theorem 1, we check the geometrical conditions of mountain pass Theorem 2 for Indeed

(Ii) since the embeddings X ^ Li(x)(Q) (i := p, r, q) and X ^ Li(

x)(dQ) (i := p, q) is are

compact, there exist positive constants C such that

|u|i(x) < Ci||u||. (3.8)

From (Go)-(Gi) it follows, for all e > 0, there exists C£ > 0, such that

G(x,u) < -^-\\u\\p+ + C£\\v\\q{x), for all (x,t) G 9Q x R, (3.9)

thus, for u € X with || u ||< 1. By (Ai), (3.2), (3.4), (3.8) and (3.9), we have

/a(u) >

min(L, 1)

+ ACr |/i|Lp(x)/(p(x)-r(x))/Q)

M--"- u„

r> iiuiip

Ci - AC2||u|r+-p+ - C3||u||q+-p+

min(L, 1) eC£Cp - -~---ZI-& ü2 Cr 1 f11 lp(

X)/(P(X)-r(X))(Q)

C3 — Cq Cg •

If p — ||u||, we obtain

Ia(u) > pp+ [Ci - AC2pr+ -p+ - C3pq+-p+

A straightforward computation shows that the maximum of the function ^ is

_ (q+(p+ ~ r+)XC2\\ Pm~ v r+(q+-r+)C3 JInserting this into equation (3.11), we find that the right side is zero for

\\*__nQ+ — r+ __—r+

& rm I m

So, there exist p > 0 and q > 0 such that Ia(w) > Q for ||u|| = p, from which the demonstration of (Ii) is completed. Now, put

h(r)= t-^G(x,ri) - G(x,t) Vt > 1.

h& (t) — (g(x, ir)ir - G(x, ir)) > 0 Vi > 1

by (G2). Hence, h(r) > h(1) for all t > 1 that is,

G(x, tî) > tt) V(x, t) € dQ x R. Let u € X, for t > 1, by (Aq) and (3.12), we have

h{tu)= [ -j—A{\\Vtu\\p{x))dx + [ —j—b(x)\\tu\\p^x\\laT — A [ F{x,tu)dx - [ G(x,tu)dar Jn P(x) Jon P(x) Jn Jon

< tp r+ A

/ -—A(|VM|p(:E))£te+ / —-(x)Mp(-TWT &n p(x) ./an p(x)

[ fi(x)\\u\\r{x)dx-C4iïl [ ^-\\v\\p+ +C£\\v\\q[x) r+ifl Jan Lp+

This shows that /a(tu) < 0.

Since /a(0) — 0, the mountain pass lemma implies the existence of a nontrivial weak solution u1 with /A(u1) >

Hence problem (1.1) has at least one nontrivial weak solution in X.

To complete the proof of the Theorem 1, one must check the conditions of the Theorem 3. So we need some lemmas which we recall below.

Remark 2. [30] As the Sobolev space X is a reflexive and separable Banach space, there exist (en)neN* C X and (/n)ragN* C X* such that /n(em) = for any n, m € N* and

X — span{en : n € N*}, X* — span{/n : n € N*}

For k € N* denote by Xk = span{efc}, Yk = ®kj=lXj, Zk = ®fXj.

Lemma 3. Assume that (A0)-(A{), (Fo)-(F1) and (G0) —(G1) hold. Then there exists X > 0, k € N and p, d > 0 such that /A/dBp n Xk > d for all 0 < A < X.

Proof. Similarly to (3.10), we have

r+ —

Taking p — ||u||, we get

/a(u) > ||u|^ [c1 - ac2|u| -p+] - c3|u|q

/a(u) > pP+ [C1 - AC2pr+-p+] - C3pq+

Next, we take A — C1/C2 ■ p^-r+ > 0 so that

Ia(U) > [Ci - AC2p^—p+] - Cap9+ > 0, which shows that I verifies the condition (I&i) in Theorem 3. □

Finally, to show the condition (I&2) in Theorem 3, we use the following lemma.

Lemma 4. Assume that (A0)-(Ai) and (G2)-(G3) hold. Then, given m € N, there exist a subspace W of X and a constant Mm > 0, independent of A, such that dim W = m and max !a(u) < Mm.

Proof. Let O and U be defined respectively as in (Fi) and in (G3). We can build the space

W, in the same way as in [28, Lemma 4.3]. So, we consider vi,......, such that v € Cœ(Q),

supp v n supp Vj = 0, supp v n O = 0 and supp v n U = 0, where i = 1,..., m, j = 1,..., m, i = j. By (2), we have

h(u)= I -¡—A(\\Vu\\p{x))dx + I -¡—b(x)\\u\\p{x)daT-X I F(x,u)dx- I G(x,u)daT Jn P(x) Vdn P(x) ./n Jdn

< max(||-o,||p , |H|P+) — A f F(x,u)dx — [ G(x,u)dax,

p Jn Jdn

where K is defined in (A0).

For u € W, since supp u n O = 0 we get

Ix(u) < max(^&A) max(||«||p~, \\\\u\\\\p+) - [ G(x, u)dax = I(u). p Jdn

max /a(u) < max !(u) — max /"(v).

«ew\\{0} «ew\\{0} vedBi (o)nw\\{q}

For t > 0 and u € dB1(0) n W\\{0} and e small enough, by (F1), (G2)-(G3) and (3.9), we obtain

I(tu) = max(^&A) max(||i«||p~, \\\\tu\\\\p+) - [ G(x,tu)dax P Jdn

< C5||iuf~ - f [ (^t\\u\\p+ + dax < C5tp~ ||uf ~ " C^l\\\\u\\\\q~,

where C5 = max(1, K)/p and C6 is the constant of embedding X ^ Lq(x)(dQ),

lim 7(iu) < lim [C5tp- - CaiH . (3.13)

i^+œ i^+œ L J

Since ß > p-, by (3.13) we get that there exist a subspace W of X and a constant Mm > 0,

independent of A, such that dim W = m and max /A(u) < Mm. The proof of Lemma 4 is complete.□

According to Lemma 2, we also have that satisfies (I&s). Since /A(0) = 0 and /A is even, we may apply Theorem 3 to conclude that /A has infinitely many nontrivial solutions.

Acknowledgements

The authors would like to thank the referees for their valuable comments which improved the presentation of the original manuscript.

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