Multiple small solutions for p ( x ) -Schrödinger equations

Multiple small solutions for p ( x ) -Schrödinger equations

with local sublinear nonlinearities via genus theory

Rabil Ayazoglu (Mashiyev)

, Ismail Ekincioglu

and Gulizar Alisoy

1Bayburt University, Faculty of Education, Bayburt, Turkey

2Institute of Mathematics and Mechanics, Baku, Azerbaijan

3Dumlupinar University, Department of Mathematics, Kutahya, Turkey

4Namik Kemal University, Faculty of Science and Arts, Tekirdag, Turkey

Received 20 April 2017, appeared 10 November 2017 Communicated by Dimitri Mugnai

Abstract. In this paper, we deal with the followingp(x)-Schrödinger problem:

(−div(|∇u|^p(x)−2∇u) +V(x)|u|^p(x)−2u= f(x,u) inR^N; u∈W^1,p(x)(_R^N),

where the nonlinearity is sublinear. We present the existence of infinitely many solutions for the problem. The main tool used here is a variational method and Krasnoselskii’s genus theory combined with the theory of variable exponent Sobolev spaces. We also establish a Bartsch–Wang type compact embedding theorem for the variable exponent spaces.

Keywords: p(x)-Laplace operator, Schrödinger equation, variable exponent Lebesgue–

Sobolev spaces, Krasnoselskii’s genus.

2010 Mathematics Subject Classification: 35J35, 35J60, 46E35.

1 Introduction

In this paper, we consider the following p(x)-Schrödinger equation inR^N (−div(|∇u|^p(x)−2∇u) +V(x)|u|^p(x)−2u= f(x,u), inR^N ;

u∈W^1,p(x)(_R^N), (P)

where N ≥ 2, p : R^N → _R is Lipschitz continuous, 1 < p⁻ := inf_x∈RNp(x) ≤ p⁺ := sup_x∈RN p(x)< NandV ∈C(_R^N,R)is the new potential function, f obeys some conditions which will be stated later andW^1,p(x)(_R^N)is the variable exponent Sobolev space.

BCorresponding author. Email: rabilmashiyev@gmail.com

These interests are stimulated mainly by the development of the studies of problems in elasticity, electrorheological fluids, flow in porous media, calculus of variations, differential equations with p(x)-growth (see [1,4,10,20,27,28]). We refer to the p(x)-Laplace operator

∆p(x)u:=div(|∇u|^p(x)−2∇u), where pis a continuous non-constant function. This differential operator is a natural generalization of thep-Laplace operator∆pu:=div(|∇u|^p−2∇u), where p > 1 is a real constant. However, the p(x)-Laplace operator possesses more complicated nonlinearity than p-Laplace operator, due to the fact that ∆p(x) is not homogeneous. This fact implies some difficulties; for example, we cannot use the Lagrange multiplier theorem in many problems involving this operator. Among these problems, the study involving p(x)- Laplacian problems via variational methods is an interesting topic. Many researchers have devoted their work to this area (see [2,3,13,15,17,26]).

WhenV(x)is radial (for exampleV(x)≡1), Dai studied the following problem in [9]:

(−div(|∇u|^p(x)−2∇u) +|u|^p(x)−2u= f(x,u), inR^N ; u∈W^1,p(x)(_R^N),

by means of a direct variational approach and the theory of variable exponent Sobolev spaces, sufficient conditions ensuring the existence of infinitely many distinct homoclinic radially symmetric solutions is proved.

The case of p, when V are radially symmetric on R^N with 1 < p⁻ ≤ p⁺ < N and V⁻ > 1 was discussed by Ge, Zhou and Xue in [19]. The existence of at least two nontrivial solutions has been established. In [30], Zhou and Wang studied the existence of infinitely many solutions for a class of (P) when the potential function does not satisfy the coercive condition.

For p(x) = p, problem (P) reduces to

(−_∆pu+V(x)|u|^p−2u= f(x,u), inR^N; u∈W^1,p R^N

. (P0)

Liu [24] studied that the existence of ground states of problem (P₀) with a potential which is periodic or has a bounded potential. Liu, Wang [23] discussed the problem (P₀) with sign-changing potential and subcriticalp-superlinear nonlinearity, by using the cohomological linking method for cones, and obtained an existence result of nontrivial solution. Recently, Alves, Liu [3] established the existence of ground state solution for problem (P) via modern variational methods on the potential functionV under following hypothesis

(V₀) V∈C(_R^N,R), inf

x∈_R^NV(x)>0

and for each M > 0, µ{x ∈ _R^N : V⁻¹(−_∞,M]} < +_{∞, where} µ(.) denotes the Lebesgue measure inR^N. By using a variational method combined with the theory of variable exponent Sobolev spaces, Duan, Huang [13] and Wang, Yao, Liu [29] studied the existence of infinitely many solutions for a class of (P) equations inR^N, which the potential V satisfies hypothesis (V0)and f(x,u)is sublinear at infinity inu. Moreover, the authors proposed new assumptions on the nonlinear term to yield bounded Palais–Smale sequences and then proved that the special sequences converge to critical points respectively. The main arguments are based on the geometry supplied by the fountain theorem. Consequently, they showed that the problem (P) under investigation admits a sequence of weak solutions with high energies.

Now, p(x)_andV(x)satisfy the following assumptions.

(p₁) The function p:R^N →_Ris ipschitz continuous and 1< p⁻ ≤ p⁺ <N;

(V₁) V ∈C R^N,R

, inf_x∈RNV(x)>0 and there are constantsr>0,α> Nsuch that for any b>0,

|y|→+lim_∞µ

x∈ Br(y): V(x)

|x|^α ≤b

=0,

where B_r(y) =x∈_R^N :|x−y|<r andµ(·)denotes the Lebesgue measure.

Remark 1.1. Condition (V₁) which is weaker than (V₀), is originally introduced by Bartsch, Wang, Willem [6] to guarantee the compact embedding of the working space. There are functionsVsatisfying(V₁)and not satisfying(V0).

In the present paper, we concern with the existence of infinitely many solutions for (P) in R^N without any growth conditions imposed on f(x,u)at infinity with respect touand to the best of our knowledge, no results on this case have been obtained up to now. The main tool used here is a variational method and Krasnoselskii’s genus theory combined with the theory of variable exponent Sobolev spaces. We also prove a Bartsch–Wang type compact embedding theorem for variable exponent spaces. We emphasize that in our approach, no coerciveness hypothesis (V₁)and not necessarily radially symmetric will be required on the potentialV . Based on the above fact and motivated by techniques used in [24,29,30], the main purpose of this paper is devoted to investigate the existence of infinitely many solutions for problem (P) when the nonlinearity is sublinear inuat infinity.

Assume that f :R^N×_R→_Rsatisfies Carathéodory and the following conditions.

(f₁) |f(x,t)| ≤m(x)g(x)|t|^m(x)−1, ∀(x,t)∈_R^N×_R, where g:R^N →_R⁺is a positive continuous function such thatg∈ L

q(x)

q(x)−m(x) R^N

∩L^∞ R^N , m∈ C R^N

, 1 < m⁻ ≤ m⁺ < p⁻, p(x) < q(x) p^∗(x) = _N^{N p}_−p⁽₍^x_x⁾₎, and q(x) p^∗(x)means that ess inf_x∈RN(p^∗(x)−q(x))>0.

(f₂) There exist an x₀∈ _R^N and a constantr>0 such that lim inf

t→0 inf

x∈Br(x₀)

Rt

0 f(x,s)ds

|t|^p(x)

!

>−_∞,

lim sup

t→0

x∈infBr(x0)

Rt

0 f(x,s)ds

|t|^p−

!

= +_∞.

(f₃) f is an odd function according to t, that is,

f(x,t) =−f(x,−t) for allt ∈_Rand for allx∈_R^N.

2 Preliminaries

In this section, we recall some results on variable exponent Lebesgue and Sobolev spaces. Over the last decade, the variable exponent Lebesgue spaces L^p(x) and the corresponding Sobolev

spaceW^1,p(x) have been a subject of active research area (we refer to [11,14,16,18,21,25] for the fundamental properties of these spaces). Write

C+(_R^N) =ⁿ_p∈C(_R^N)_{: p}(_x)>_{1 for any x}∈_R^No, p⁻ := inf

x∈_R^Np(x),p⁺ := sup

x∈_R^N

p(x) for any p∈C+(_R^N).

The set of all measurable real-valued functions defined onR^N will be denote by<(_R^N). Note that two measurable functions in<(_R^N)are considered as the same element of<(_R^N)when they are equal almost everywhere.

Let p∈C+(_R^N). The variable exponent Lebesgue space L^p(x)(_R^N)is defined by L^p(x)(_R^N) =

u∈ <(_R^N): Z

R^N|u(x)|^p(x)dx<_∞

, which is equipped with the norm, so-called Luxemburg norm

|u|_Lp(x)(_R^N):= |u|_p(x) =inf (

µ>0 : Z

R^N

u(x) µ

p(x)

dx ≤1 )

. Moreover, we define the variable exponent Sobolev space by

W^1,p(x)(_R^N) =u∈ L^p(x)(_R^N):|∇u| ∈ L^p(x)(_R^N) , with the norm

kuk_W1,p(x)(_R^N):=kuk_1,p(x)= |u|_p(x)+|∇u|_p(x)

for allu∈W^1,p(x)(_R^N).The spacesL^p(x)(_R^N),W^1,p(x)(_R^N)are separable and reflexive Banach spaces [11,18,21]. Now, let us introduce the modular of the spaceL^p(x) R^N

as the functional σ_p(x) :L^p(x)(_R^N)→_Rdefined by

σ_p(x)(u) =

Z

R^N|u(x)|^p(x)dx

for allu∈ L^p(x)(_R^N). The relation between modular and Luxemburg norm is clarified by the following propositions.

Proposition 2.1([11,18,21]). Let u,un∈ L^p(x)(_R^N) (n=1, 2, . . .): (i) |u|_p(x)<₁(=_1;>₁)⇔σ_p(x)(u)<₁(=_1;>₁)_;

(ii) |u|_p(x)>₁ =⇒ |u|^p_p⁻_(x) ≤σ_p(x)(u)≤ |u|^p_p⁺_(x);

|u|_p(x)<1 =⇒ |u|^p_p⁺_(x) ≤σ_p(x)(u)≤ |u|^p_p⁻_(x); (iii) |u|_p(x)≤σ_p(x)(u)^p1− +σ_p(x)(u)^p1+ ;

(iv) lim

n→_∞|u_n−u|_p(x)=0⇔ lim

n→_∞σ_p(x)(u_n−u) =0;

nlim→_∞|u_n|_p(x) =_∞⇔σ_p(x)(u_n) =_∞.

Proposition 2.2([11,21], Hölder-type inequality). The conjugate space of L^p(x)(_R^N)is L^p0(x)(_R^N), where _p(¹_x)+ _p0¹(x) =1. For any u ∈L^p(x)(_R^N)and v ∈L^p0(x)(_R^N), we have

Z

R^Nuvdx

≤2|u|_p(x)|v|_p0(x).

Proposition 2.3 ([2,11,16]). Let k,h ∈ C(_R^N) with 1 < k(x) ≤ h(x) for all x ∈ _R^N and u ∈ L^h(x)(_R^N). Then,|u|^k(x)∈ L^h

(x)

k(x)(_R^N)via

|u|^k(x)_h(x)

k(x)

≤ |u|^k_h⁻_(x)+|u|^k_h⁺_(x)

or there exists a numberbk∈ [k⁻,k⁺]such that

|u|^k(x)_h(x)

k(x)

=|u|^bk_h(x).

Proposition 2.4 ([14,16,21]). Let p : R^N → _R be Lipschitz continuous and satisfy p⁺ < N, let q:R^N →_Rbe a measurable function. If p(x)≤ q(x)≤ p^∗(x) = ^{N p(x)}

N−p(x), then there is a continuous embedding W^1,p(x)(_R^N),→ L^q(x)(_R^N).

Proposition 2.5 ([10,15,19]). Let Ω be a bounded domain in R^N. Assume that the boundary ∂Ω possesses cone property and q(x) ∈ C(_Ω,R)with1 ≤ q(x) p^∗(x) = _N^{N p}_−p^(x_(x⁾₎ for N > p(x)and p^∗(x) = +_∞for N ≤ p(x), then there is a compact embedding W^1,p(x)(_Ω),→,→ L^q(x)(_Ω).

Proposition 2.6([18,21]). LetΩbe an open subset ofR^N and G:Ω×_R→_Rsatisfies Carathéodory conditions, and

|G(x,t)| ≤a(x) +b|t|^p1(x)/p2(x), ∀(x,t)∈_Ω×_R,

where a ∈L^p2(x)(_Ω), b is a positive constant, p₁,p₂∈ L^∞(_Ω). Denoted by N_Gthe Nemytsky operator is defined by G, i.e.

(N_G(u))(x) =G(x,u(x)),

then N_F : L^p1(x)(_Ω) → L^p2(x)(_Ω) is a continuous and bounded map. When p(x) < N, write p^∗(x) = _N^{N p}_−p^(x_(x⁾₎.

Next, we consider the case thatV satisfies(V₁). On the linear subspace E=

u∈W^1,p(x) R^N

: Z

R^N

|∇u|^p(x)+V(x)|u|^p(x)dx<+_∞

, we equip with the norm

kuk_E =inf (

η>0 : Z

R^N

∇u η

p(x)

+V(x) u η

p(x)!

dx≤1 )

.

Then (E,k·k_E)is continuously embedded into W^1,p(x) R^N

as a closed subspace. Therefore, (E,k·k_E)is also a separable reflexive Banach space.

In addition, we define the modularΛ_p(x),V :E→_Rassociated with Eas follows:

Λ_p(x),V(u) =

Z

R^N

|∇u(x)|^p(x)+V(x)|u(x)|^p(x)dx

for all u∈E, in a similar way to Proposition2.1. The following proposition holds.

Proposition 2.7([11,18,21]). Let u,u_n∈ E:

(i) kuk_E <1(=1;>1)⇔_Λp(x),V(u)<1(=1;>1); (ii) kuk_E >1 =⇒ kuk^p_E⁻ ≤_Λp(x),V(u)≤ kuk_E^p+;

kuk_E <1 =⇒ kuk^p_E⁺ ≤_Λp(x),V(u)≤ kuk^p_E⁻; (iii) lim

n→_∞ku_nk_E =0⇔ lim

n→_∞Λ_p(x),V(u_n) =0;

nlim→_∞kunk_E = _∞⇔ lim

n→_∞Λ_p(x),V(un) =_∞.

Proposition 2.8([17]). J ∈C¹(E,R)and J⁰(u),υ

=

Z

R^N

|∇u|^p(x)−2∇u∇υ+V(x)|u|^p(x)−2uυ

dx, ∀υ,u∈ E,

J is a convex functional, J⁰ : E → E^∗ is strictly monotone, bounded homeomorphism, and is of (S+) type, namely un*u (weakly) and

limn→_∞

J⁰(un)−J⁰(u),(un−u)≤0 implies un→u (strongly) in E.

Definition 2.9. We say that the functional J satisfies the Palais–Smale condition ((PS) for short) if every sequence{u_n} ∈Esuch that

|I(u_n)| ≤c and I⁰(u_n)→0 asn→_∞ contains a convergent subsequence in the norm ofE.

3 Proof of main result

In order to discuss the problem (P), we need to consider the energy functional I : E → _R defined by

I(u) =

Z

R^N

1 p(x)

|∇u|^p(x)+V(x)|u|^p(x)dx−

Z

R^NF(x,u)dx,∀u∈ E, (3.1) whereF(x,t) =Rt

0 f(x,s)ds,∀(x,t)∈_R^N×_R. Set J(u) =

Z

R^N

1 p(x)

|∇u|^p(x)+V(x)|u|^p(x)dx, Ψ(u) =

Z

R^NF(x,u)dx, then

I(u) =J(u)−_Ψ(u).

Under our conditions, it follows from Hölder-type inequality and Sobolev embedding theorem that the energy functional I is well-defined. It is well known that I ∈ C¹(E,R)and its derivative is given by

I⁰(u),ϕ

=

Z

R^N

|∇u|^p(x)−2∇u∇ϕ+V(x)|u|^p(x)−2uϕ dx−

Z

R^N f(x,u)ϕdx (3.2) for allu,ϕ∈E.

Definition 3.1. We call thatu∈E\ {0}is a weak solution of (P), if Z

R^N

|∇u|^p(x)−2∇u∇ϕ+V(x)|u|^p(x)−2uϕ dx=

Z

R^N f(x,u)ϕdx, where ϕ∈E.

We are now in a position to state our main results.

Theorem 3.2. Assume that conditions(p₁),(V₁)and(f₁)hold.

(1) Problem(P)has a solution.

(₂) Furthermore, if f admits the conditions(_f2)_and(_f3), then problem(P)has a sequence of solution {±u_k :k=1, 2, . . . ,}such that I(±u_k)<0and I(±u_k)→0as k→_∞.

For more existence and multiplicity results on p(x)-Laplacian equation inR^N, we refer to Fan and Han [15]. In [15, Theorem 3.2], the similar results are obtained for the caseV(x)≡1.

For the proof of Theorem3.2 we need some preliminary lemmas. The following Bartsch–

Wang type compact embedding will play a crucial role in our subsequent arguments.

Lemma 3.3. If V satisfies(V₁)and p:R^N →_Ris continuous and1< p⁻ ≤ p⁺ <N, then (i) we have a compact embedding E,→,→ L

p(x) p− (_R^N).

(ii) for any measurable function q : R^N → _R with ^p_p⁽−^x) < q(x) p^∗(x), we have a compact embedding E,→,→ L^q(x)(_R^N).

Proof. (i). Let {u_n} ⊂ Esuch that ku_nk_E ≤ C. We assume up to a subsequence u_n * uin E.

Then, we havekωnk_E ≤ Candωn*0 in E, whereωn =un−u. We need to show ωn→0 in L

p(x)

p− (_R^N)to complete the proof.

SetA_b(y) =x ∈B_r(y): ^V_|x⁽_|^xα⁾ ≤b ∩B_r(y_i)andD_b(y) =x∈ B_r(y): ^V_|x⁽_|^xα⁾ >b ∩B_r(y_i). By the Sobolev compact imbedding theorem in bounded domains (see Proposition 2.5), it implies that ω_n → 0 strongly in L

p(x)

p− (B_R) for any R > 0, where B_R = x ∈_R^N :|x| ≤R . To estimate R

B^c_R|ω_n|

p(x) p−

dx, let {y_i}_i∈N be a sequence of points in R^N satisfying such that R^N ⊂ ∪^∞_i=1B_r(y_i) and each point x is contained in at most 2^N such balls B_r(y_i). So for all R>2r, we have

Z

B_R^c

|ωn|

p(x) p−

dx≤

∑

|y_i|≥R−r

Z

Br(yi)

|ωn|

p(x) p−

dx

=

∑

|y_i|≥R−r

Z

A_b(yi)

|ω_n|

p(x) p−

dx+

∑

|y_i|≥R−r

Z

D_b(yi)

|ω_n|

p(x) p−

dx.

On the other hand, choose a numberτ∈ 1,_N−^N_p−

arbitrarily, then p(x)<τp(x)< p^∗(x). By Proposition2.4, we have that

E,→W^1,p(x)(_R^N),→L^τp(x)(_R^N) is continuous, and there are constantsC₀,C₁,C>0 such that

|ω_n|_τp(x) ≤C₀kω_nk_1,p(x) ≤C₁kω_nk_E ≤C₁C, ∀ω_n ∈E. (3.3)

Applying the Hölder-type inequality, we get

|y_i|≥

∑

Z

A_b(y_i)

|ω_n|

p(x) p−

dx≤

∑

|y_i|≥R−r

|ω_n|

p(x) p−

τp⁻,A_b(y_i)

|1| _τp− τp− −1,Ab(yi)

≤

∑

|y_i|≥R−r

|ω_n|_τp(x),A

b(yi)+|ω_n|

p+ p−

τp(x),A_b(y_i)

sup

|yi|≥R−r

[µ(A_b(y_i))]

τp− −1 τp−

≤

∑

|yi|≥R−r

|ωn|_τp(x),B

r(yi)+|ωn|

p+ p−

τp(x),Br(y_i)

sup

|y_i|≥R−r

[µ(A_b(y_i))]

τp− −1 τp−

≤2^N

|ω_n|_τp(x),Bc

R−2r+|ω_n|

p+ p− τp(x),B^c_R−2r

sup

|y_i|≥R−r

[µ(A_b(y_i))]

τp− −1 τp−

≤2^N

|ωn|_τp(x)+|ωn|

p+ p− τp(x)

sup

|y_i|≥R−r

[µ(A_b(y_i))]

τp− −1 τp−

≤2^NC

bp p− 1 kωnk

bp p−

E sup

|yi|≥R−r

[µ(A_b(y_i))]

τp− −1 τp−

≤2^N(C₁C)

bp p−

sup

|yi|≥R−r

[µ(A_b(y_i))]

τp− −1 τp−

,

where _p^pb− ∈1, ^p_p⁺−

and|u|_s(x),Ω =inf

ν>0 :R

Ω

^u

(x) ν

s(x)

dx ≤1 . SettingΩn:={x∈_R^N :|x|^α|ωn|

_{p− −1}

p−

p(x)

≤1}, we conclude that

|y_i|≥

∑

Z

D_b(y_i)

|ω_n|

p(x) p−

dx

=

∑

|yi|≥R−r

Z

D_b(y_i)∩_Ω^c_n|ω_n|

p(x) p−

dx+

∑

|yi|≥R−r

Z

D_b(y_i)∩_Ωn

|ω_n|

p(x) p−

dx

≤

∑

|yi|≥R−r

Z

Db(yi)∩_Ω^c_n|x|^α|ωn|

p(x)

p− |x|^−αdx+

∑

|yi|≥R−r

Z

Db(yi)∩_Ωn|x|^{p− −−α1} dx

≤

∑

|yi|≥R−r

Z

D_b(y_i)

|x|^α|ω_n|^p(x)dx+

∑

|yi|≥R−r

Z

D_b(y_i)

|x|^{p− −−α1} dx

≤

∑

|yi|≥R−r

Z

Db(yi)

|x|^α|ωn|^p(x)dx+

∑

|yi|≥R−r

Z

Db(yi)

|x|^{p− −−α1} dx

≤ ¹

b

∑

|yi|≥R−r

Z

D_b(y_i)V(x)|ω_n|^p(x)dx+

∑

|yi|≥R−r

Z

D_b(y_i)|x|

−α p− −1 dx

≤ ²

N

b Z

B^c_R−2rV(x)|ωn|^p(x)dx+

Z

B^c_R−2r

|x|^{p− −−α1} dx

≤ ²

N

b Z

R^N(|∇ω_n|^p(x)+_V(_x)|ω_n|^p(x))_dx+ ²

N

(R−2r)^{p− −α1−N}

≤ ²

N

b kωnk^p_E^b+ ²

N

(R−2r)^{p− −α1−N}

≤ ²

NC^bp

b + ²

N

(R−2r)^{p− −α1−N} withα> _p−^N−1. Therefore, we get

Z

B^c_R|ω_n|

p(x) p−

dx ≤2^N(C₁C)

bp p−

sup

|yi|≥R−r

[µ(A_b(y_i))]

τp− −1 τp− + ²

NC^bp

b + ²

N

(R−2r)

α

p− −1−N. (3.4) Now, for anyε >0 and choose b> 0, we can find a positive constant R>0 big enough such that

2^N(C₁C)

bp p−

sup

|y_i|≥R−r

[µ(A_b(y_i))]

τp− −1 τp− < ^ε

6, (3.5)

2^NC^pb b < ^ε

6, (3.6)

and

2^N (R−2r)^{p− −α1−N}

< ^ε

6. (3.7)

It follows from (3.4)–(3.7) that

Z

B^c_R

|ω_n|

p(x) p−

dx≤ ^ε 2. This implies ω_n→0 in L

p(x)

p− (_R^N)and the proof of Lemma3.3(i)is completed.

(ii). Since {ωn} is bounded in E and by Proposition 2.4, we get that {ωn} is bounded in L^p∗(x)(_R^N). It is taken into account ^p_p⁽−^x) < q(x) p^∗(x), we can use the interpolation inequality [21, Corollary 2.2] and obtain

|ω_n|_q(x) ≤c|ω_n|^η_p(x)

p−

|ω_n|^σ_p∗(x), ∀ω_n∈ L

p(x) p−

(_R^N)∩L^p∗(x)(_R^N), (3.8) wherec>0 is a constant and

η=











ess sup

x∈_R^N

p(x) q(x)

p^∗(x)−q(x)

p⁻p^∗(x)−p(x) ^if |ωn|p(x) p− >1, ess inf

x∈_R^N

p(x) q(x)

p^∗(x)−q(x)

p⁻p^∗(x)−p(x) ^if |ωn|p(x) p− ≤1, and

σ=









 ess sup

x∈_R^N

p^∗(x) q(x)

p⁻q(x)−p(x)

p⁻p^∗(x)−p(x) ^if |ωn|_p∗(x) >1, ess inf p^∗(x)

q(x)

p⁻q(x)−p(x)

p⁻p^∗(x)−p(x) ^if |ω_n|_p∗(x) ≤1.

Moreover, it follows from Lemma 3.3 (i) that |ωn|p(x)

p− → 0 and (3.8) implies ωn → 0 in L^q(x)(_R^N). In the proof, some ideas in [5,6,12] have been followed. Proof of Lemma3.3 is completed.

Lemma 3.4. Suppose(f₁)holds. Then functional I is coercive and bounded from below.

Proof. Let M = |g| q(x) q(x)−m(x)

. From (f₁), we have |F(x,t)| ≤ g(x)|u|^m(x). By using Proposi- tions2.1,2.2,2.3,2.4,2.7, we get

I(u)≥ ¹ p⁺

Z

R^N

|∇u|^p(x)+V(x)|u|^p(x)dx−

Z

R^Ng(x)|u|^m(x)dx

≥ ¹

p⁺kuk^p_E⁻−2|g| q(x) q(x)−m(x)

|u|^m_q^b_(x)

≥ ¹

p⁺kuk^p_E⁻−2C^m₁^bMkuk^m_E^b, ∀u∈ E, (3.9) for kuk_E large enough. Therefore, mb < p⁻ gives the coercivity of I and I is bounded from below. Proof of Lemma3.4is completed.

Lemma 3.5. Suppose(f₁)holds. Then I satisfies the(PS)condition.

Proof. Let us assume that there exists a sequence{u_n}in Esuch that

|I(u_n)| →c and I⁰(u_n)→0 as n→_∞. (3.10) From (3.10), we have |I(u_n)| ≤c₀. Combining this fact with (3.9) implies that

c0 ≥ I(un)≥ ¹

p⁺kunk_E^p−−2C₁^mb|g| q(x) q(x)−m(x)

kunk^m_E^b ≥c₁,

for kuk_E large enough. Since mb < p⁻, we obtain that {u_n} is bounded in E. Finally, we show that there is a strongly convergent subsequence of {un} in E. Indeed, in view of the boundedness of{u_n}, passing to a subsequence if necessary, still denoted by {u_n}, we may assume that

un *u in E.

By Lemma3.3, we obtain the following results:

u_n→u in L

p(x)

p− (_R^N), p(x)

p⁻ ≤q(x) p^∗(x), that is

|u_n−u|p(x) p−

→_{0 as}n→_∞, and

|u_n−u|_q(x)→0 as n→_∞.

In view of the definition of weak convergence, we havehI⁰(u_n)−I⁰(u),u_n−ui →0. Hence J⁰(u_n)−J⁰(u),u_n−u

= I⁰(u_n)−I⁰(u),u_n−u +

Z

R^N(f(x,u_n)− f(x,u)) (u_n−u)dx

=

Z

R^N|∇un|^p(x)−2∇un(∇un− ∇u)dx +

Z

R^NV(x)|u_n|^p(x)−2u_n(u_n−u)dx +

Z

R^N(f(x,u_n)− f(x,u))(u_n−u)dx→0. (3.11) It is clear that

I⁰(u_n)−I⁰(u)_,u_n−u

=I⁰(u_n)_,u_n−u

+I⁰(u)_,u_n−u

→_{0. (3.12)}