Existence of solutions for perturbed fourth order elliptic equations with variable exponents

Existence of solutions for perturbed fourth order elliptic equations with variable exponents

Nguyen Thanh Chung

Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam Received 20 June 2018, appeared 4 December 2018

Communicated by Gabriele Bonanno

Abstract. Using variational methods, we study the existence and multiplicity of solu- tions for a class of fourth order elliptic equations of the form

(∆²_p(x)u−M R

Ω 1

p(x)|∇u|^p(x)dx

∆_p(x)u= f(x,u) inΩ, u=_∆u=0 on∂Ω,

where Ω ⊂ R^N, N ≥ 3, is a smooth bounded domain, ∆²_p(x)u = ∆(|∆u|^p(x)−2∆u) is the operator of fourth order called the p(x)-biharmonic operator, ∆p(x)u = div |∇u|^p(x)−2∇u

is thep(x)-Laplacian, p :Ω →Ris a log-Hölder continuous func- tion, M : [0,+∞) → R and f : Ω×R → R are two continuous functions satisfying some certain conditions.

Keywords:fourth order elliptic equations, Kirchhoff type problems, variable exponents, variational methods.

2010 Mathematics Subject Classification: 35J60, 35J35, 35G30, 46E35.

1 Introduction

In this paper, we are interested in the existence of weak solutions for the following fourth order elliptic equations







∆²_p(x)u−M R

Ω 1

p(x)|∇u|^p(x)dx

∆_p(x)u= f(x,u) inΩ,

u= _∆u=0 on ∂Ω, (1.1)

where Ω ⊂ _R^N, N ≥ 3, is a smooth bounded domain, ∆²_p(x)u = _∆ |_∆u|^p(x)−2_∆u is the operator of fourth order called the p(x)-biharmonic operator,∆_p(x)u=div |∇u|^p(x)−2∇u

is the p(x)-Laplacian, the exponent p : Ω → _R is log-Hölder continuous, that is, there exists c> 0 such that|p(x)−p(y)| ≤ −_log|^c_x−y| for all x,y ∈_Ωwith 0< |x−y| ≤ ¹₂ and 1< p⁻ :=

BCorresponding author. Email: ntchung82@yahoo.com

inf_x∈Ωp(x)≤ p⁺:=sup_x∈Ωp(x)< ^N₂, the function M ∈C([0,+_∞),R)may be degenerate at zero and f : Ω×_R→ _Ris a continuous function satisfying the following subcritical growth condition:

(F₀) There existsC>0 such that

|f(x,t)| ≤C(1+|t|^q(x)−1), ∀x ∈_Ω, t∈ _R, whereq∈C+(_Ω)andq(x)< p^∗₂(x) = _N^{N p}_−2p^(x₍⁾_x) for all x∈_Ω.

We point out that if p(.)is a constant then problem (1.1) has been studied by many authors in recent years, we refer to some interesting papers [4,11,21,26,27,31,32,36–38,40]. In [38], Wang and An considered the following fourth-order elliptic equation

(∆²u−M R

Ω|∇u|²dx

∆u= f(x,u) inΩ,

u=_∆u=_{0 on} ∂Ω, (1.2)

whereΩ⊂ _R^N_, N ≥ 1, is a smooth bounded domain, M : [_0,+_∞)→_{R and} f : Ω×_R→ _R are two continuous functions. This problem is related to the stationary analog of the evolution equation of Kirchhoff type

u_tt+_∆²u−M Z

Ω|∇u|²dx

∆u= f(x,t)_{, (1.3)}

where ∆² is the biharmonic operator, ∇u denotes the spatial gradient of u, see [8] for the meaning of the problem from the point of view of physics and engineering. By assuming that M is bounded on [0,+_∞)and the nonlinear term f satisfies the Ambrosetti–Rabinowitz type condition, Wang et al. obtained in [38] at least one nontrivial solution for problem (1.2) using the mountain pass theorem. Moreover, the authors also showed the existence at least two solutions in the case when f is asymptotically linear at infinity. After that, Wang et al.

[37] studied problem (1.2) in the case when M is unbounded function, i.e. M(t) = a+bt, wherea >0,b≥0 by using the mountain pass techniques and the truncation method. Some extensions regarding these results can be found in [4,11,21,31,36,40] in which the authors considered problem (1.2) inR^Nor the nonlinearities involved critical exponents. In [26,27,32], problem (1.1) was studied in the general case when p(.) = p∈(1,+_∞)is a constant.

In recent years, the study of differential equations and variational problems with nonstan- dardp(x)-growth conditions has received more and more interest. The reason of such interest starts from the study of the role played by their applications in mathematical modelling of non-Newtonian fluids, in particular, the electrorheological fluids and of other phenomena re- lated to image processing, elasticity and the flow in porous media, we refer the readers to [5,35,43] for more details. Some results on problems involving p(x)-Laplace operator orp(x)- biharmonic operator can be found in [6,7,9,10,12,16,17,28,30,33]. These types of operators where p(.) is a continuous function possess more complicated properties than the constant cases, mainly due to the fact that they are not homogeneous. We also find that Kirchhoff type problems with variable exponents has received a lot of attention in recent years, see for example [1,2,13–15,18–20,41].

Motivated by the contributions cited above, in this paper we study the existence and mul- tiplicity of solutions for perturbed fourth order elliptic equations with variable exponents of the form (1.1). More precisely, we consider problem (1.1) in two case when f is sublinear or

superlinear at infinity. In the sublinear case, we obtain an existence result using the minimum principle while in the superlinear case we prove some existence and multiplicity results with the help of the Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem. To the best of our knowlegde, the present paper is the first contribution to the study of this type of problems in Sobolev spaces with variable exponents.

2 Preliminaries

We recall in what follows some definitions and basic properties of the generalized Lebesgue- Sobolev spaces L^p(x)(_Ω)and W^k,p(x)(_Ω) where Ωis an open subset of R^N. In that context, we refer to the books of Diening et al. [22] and Musielak [34], the papers of Fan et al. [24,25], Zang et al. [42], Ayoujil et al. [6,7] and Boureanu et al. [10]. Set

C+(_Ω):={h; h∈ C(_Ω), h(x)>1 for all x∈_Ω}. For anyh ∈C+(_Ω)we define

h⁺ =sup

x∈_Ω

h(x) and h⁻= inf

x∈_Ωh(x). For any p(x)∈ C+(_Ω), we define the variable exponent Lebesgue space

L^p(x)(_Ω) =

u: a measurable real-valued function such that Z

Ω|u(x)|^p(x)dx <_∞

. We recall the following so-calledLuxemburg normon this space defined by the formula

|u|_p(x)=_inf (

µ>_{0 :}

Z

Ω

u(x) µ

p(x)

dx≤₁ )

.

Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1< p⁻≤ p⁺<

∞and continuous functions are dense ifp⁺< _∞. The inclusion between Lebesgue spaces also generalizes naturally: if 0< |_Ω|<_∞ andp₁,p2 are variable exponents so that p₁(x)≤ p2(x) a.e. x ∈ _Ωthen there exists the continuous embedding L^p2(x)(_Ω),→ L^p1(x)(_Ω). We denote by L^p0(x)(_Ω)the conjugate space of L^p(x)(_Ω), where _p(¹_x) + ¹

p⁰(x) = 1. For any u ∈ L^p(x)(_Ω)and v∈L^p0(x)(_Ω)the Hölder inequality

Z

Ωuv dx

≤ 1

p⁻ + ¹ (p⁰)⁻

|u|_p(x)|v|_p0(x)

holds true.

An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by themodularof theL^p(x)(_Ω)space, which is the mappingρ_p(x) :L^p(x)(_Ω)→_Rdefined by

ρ_p(x)(u) =

Z

Ω|u|^p(x)dx.

Ifu∈ L^p(x)(_Ω)andp⁺<_∞then the following relations hold

|u|^p−

p(x)≤ρ_p(x)(u)≤ |u|^p+

p(x) (2.1)

provided|u|_p(x) >1 while

|u|^p+

p(x)≤ ρ_p(x)(u)≤ |u|^p−

p(x) (2.2)

provided|u|_p(x) <_{1 and}

|u_n−u|_p(x) →0 ⇔ ρ_p(x)(u_n−u)→0. (2.3) As in the constant case, for any positive integerk, the Sobolev space with variable exponent W^k,p(x)(_Ω)is defined by

W^k,p(x)(_Ω) ={u∈ L^p(x)(_Ω): D^αu∈ L^p(x)(_Ω), |α| ≤k}, where D^αu = ^∂|α|

∂x₁^α1∂x^α₂²...∂x^α_N^Nu, with α = (α₁, . . . ,α_N) is a multi-index and |α| = _∑^N_i=1α_i. The spaceW^k,p(x)(_Ω)equipped with the norm

kuk_k,p(x)=

∑

|α|≤k

|D^αu|_p(x),

also becomes a separable and reflexive Banach space. Due to the log-Hölder continuity of the exponent p, the space C^∞(_Ω) is dense in W^k,p(x)(_Ω). Moreover, we have the following embedding results.

Proposition 2.1 (see [24,25]). For p,r ∈ C+(_Ω) such that r(x) ≤ p^∗_k(x)for all x ∈ Ω, there is a continuous embedding

W^k,p(x)(_Ω),→L^r(x)(_Ω),

where p^∗_k(x) = _N^{N p}_−kp^(x₍⁾_x) if kp(x)< N and p^∗_k(x) = +_∞ if kp(x) > N. If we replace ≤with<, the embedding is compact.

We denote byW₀^k,p(x)(_Ω)the closure ofC₀^∞(_Ω)inW^k,p(x)(_Ω). Note that the weak solutions of problem (1.1) are considered in the generalized Sobolev space

X=W₀^1,p(x)(_Ω)∩W^2,p(x)(_Ω)

equipped with the norm kuk_X = kuk_1,p(x) + kuk_2,p(x) or kuk_X = |u|_p(x) + |∇u|_p(x) +_∑α=2|D^αu|_p(x).

According to [42], the normk.k_Xis equivalent to the norm|_∆.|_p(x) in the spaceX. Conse- quently, the normsk.k_2,p(x),k.k_X and|_∆.|_p(x) are equivalent. For this reason, we can consider in the spaceX the following equivalent norms

kuk= |_∆u|_p(x)+|∇u|_p(x) and

kuk=inf (

µ>0 : Z

Ω

∆u(x) µ

p(x)

+

∇u(x) µ

p(x)!

dx≤1 )

. Let us define the functionalΛ:X →_Rby

Λ(u) =

Z

Ω

|_∆u|^p(x)+|∇u|^p(x) dx, u∈ X, (2.4) then using similar arguments as in [10, Proposition 1] we obtain the following modular-type inequalities.

Proposition 2.2. For u,u_n ∈ X and the functional Λ : X → _R define as in (2.4), we have the following assertions:

(1) kuk<1(respectively=1;>1)⇐⇒ _Λ(u)<1(respectively=1;>1); (2) kuk ≤1⇒ kuk^p+ ≤_Λ(u)≤ kuk^p−;

(3) kuk ≥1⇒ kuk^p− ≤_Λ(u)≤ kuk^p+;

(₄) ku_nk →₀(respectively→_∞)⇐⇒_Λ(u_n)→₀(respectively→_∞)as n →_∞.

3 Main results

In this section, we will discuss the existence and multiplicity of weak solutions of problem (1.1). Let us denote byc_i,i=1, 2, . . . general positive constants whose value may change from line to line. We will look for weak solutions of problem (1.1) in the space X := W₀^1,p(x)(_Ω)∩ W^2,p(x)(_Ω) with the norm mentioned as in Section 2. First, let us make the definition of a weak solution of problem (1.1) as follows.

Definition 3.1. We say thatu ∈Xis a weak solution of problem (1.1) if Z

Ω|_∆u|^p(x)−2_{∆u∆v dx}+M Z

Ω

1

p(x)|∇u|^p(x)dx Z

Ω|∇u|^p(x)−2∇u∇v dx−

Z

Ω f(x,u)v dx=0 for all v∈X.

Let us define the functional J :X→_Rby J(u) =

Z

Ω

1

p(x)|_∆u|^p(x)dx+Mb _Z

Ω

1

p(x)|∇u|^p(x)dx

−

Z

ΩF(x,u)dx

=_Φ(u)−_Ψ(u)_,

(3.1)

where

Φ(u) =

Z

Ω

1

p(x)|_∆u|^p(x)dx+Mb _Z

Ω

1

p(x)|∇u|^p(x)dx

, Ψ(_u) =

Z

ΩF(_x,u)_{dx, u}∈ X

(3.2)

and Mb(t) =Rt

0 M(s)ds.

Using some simple computations, we can show that J ∈ C¹(X,R) and its derivative is given by the formula

J⁰(u)(v) =

Z

Ω|_∆u|^p(x)−2_{∆u∆v dx}+M Z

Ω

1

p(x)|∇u|^p(x)dx Z

Ω|∇u|^p(x)−2∇u∇v dx

−

Z

Ω f(x,u)v dx

for all u,v ∈ X. Thus, we will seek weak solutions of problem (1.1) as the critical points of the functional J. We first obtain an existence result for problem (1.1) in the case when f is sublinear at infinity. We also consider case when the Kirchhoff function M are allowed to be degenerate at zero.

Theorem 3.2. Assume that the condition (F₀)hold with1< q⁻ ≤ q⁺ < p⁻. Moreover, we assume that the following conditions hold:

(M₁⁰) There exist m⁰₀,t₀>0such that

M(t)≥m⁰₀, ∀t ≥t₀; (M₂⁰) There existsα>1such that

limt→0

M(t) t^α−1 =0;

(F₀⁰) There exist positive constants C₀,δ > 0 and a subset Ω0 ⊂ Ω, a function r ∈ C+(_Ω), r(x)< p(x)for all x∈Ω, such that

|F(x,t)| ≥C₀|t|^r(x) for all x∈ _Ω0and|t|<δ.

Then problem(1.1)has a nontrivial weak solution.

Proof. From(F₀), there existsc₁ >0 such that

|F(x,t)| ≤c₁

|t|+|t|^q(x), ∀x ∈_Ω, t ∈_R. (3.3) We also obtain from(M₁⁰)and(M⁰₂)that

Mb(t)≥ m⁰₀t, ∀t≥t₀, Mb(t)≤et^α, ∀t ∈(0,t_e), (3.4) where Mb(t) =Rt

0 M(s)dsandteis a positive constant depending one>0.

Fort₀ given as above, let us define the set

Xb :=ⁿu∈X : min{|∇u|^p_p⁺_(x),|∇u|^p_p⁻_(x)} ≥ p⁺t₀o .

Then it follows that Xb is a closed subspace of the reflexive Banach space X, so Xb is a reflexive Banach space too. Moreover, for anyu∈ X, we haveb

Z

Ω

1

p(x)|∇u|^p(x)dx ≥ ¹

p⁺minn

|∇u|^p+

p(x),|∇u|^p−

p(x)

o≥t₀.

By relations (3.3) and (3.4), by the Sobolev embedding, we deduce that for anyu∈ Xb with kuk>1 large enough,

J(u) =

Z

Ω

1

p(x)|_∆u|^p(x)dx+Mb _Z

Ω

1

p(x)|∇u|^p(x)dx

−

Z

ΩF(x,u)dx

≥ ¹ p⁺

Z

Ω|_∆u|^p(x)dx+ ^m

0 0

p⁺ Z

Ω|∇u|^p(x)dx−c₁ Z

Ω|u|^q(x)dx−c₁ Z

Ω|u|dx

≥ ^min{1,m⁰₀}

p⁺ kuk^p−−c₂kuk^q+−c₂kuk.

Since 1 < q⁺ < p⁻ it follows that the functional J is coercive in X. Moreover, we find thatb J is weakly lower semicontinuous in Xb and thus, J attains its infimum in Xb and there exists u₀ ∈Xb such that

J(u₀) = _inf

u∈Xb

J(u)_.

Next, we show that u₀ 6= 0 i.e. u₀ is a nontrivial weak solution of problem (1.1). Let x0 ∈ _Ω0. Since p,r ∈ C+(_Ω), we can chooseρ > 0 small enough such that Bρ(x0) ⊂ _Ω0 and p⁻₀ :=min_x∈Bρ(x0)p(x)> r₀⁺:=max_x∈Bρ(x0)r(x). Now, let us choose φ∈ C^∞₀ (_Ω)with |φ| ≤1, kφk

W^2,p(x)(Bρ(x₀))∩W₀^1,p(x)(Bρ(x₀))≤ c(ρ)and|φ|_Lr(x)(Bρ(x₀)) >0. Then, for any 0<t <δ we deduce from(F₀⁰)and (3.4) that

J(tφ) =

Z

Ω

1

p(x)|_∆(tφ)|^p(x)dx+Mb _Z

Ω

1

p(x)|∇(tφ)|^p(x)dx

−

Z

ΩF(x,tφ)dx

≤ ^t

p⁻₀

p⁻ Z

Ω|_∆φ|^p(x)dx+ ^et

αp⁻₀

(_p⁻)^α _Z

Ω|∇φ|^p(x)dx α

−C₀t^r0+ Z

Bρ(x₀)

|φ|^r(x)dx

≤ ^t

p⁻₀

p⁻ max{c(ρ)^p−0,c(ρ)^p+0}+ ^et

αp⁻₀

(p⁻)^{α max}{c(ρ)^αp−0,c(ρ)^αp+0} −C₀t^r+0 Z

Bρ(x0)

|φ|^r(x)dx.

Since r₀⁺ < p⁻₀ < αp⁻₀, we get J(t₁φ) < 0 by taking 0 < t₁ < δ small enough. Hence, J(u₀)≤ J(t₁φ)<0. Therefore,u₀ ∈Xb ⊂Xis a nontrivial critical point of J and problem (1.1) has a nontrivial weak solution.

In the next part of this paper, we will study the existence and multiplicity of weak solutions for problem (1.1) in the case when f is superlinear at infinity. In the sequel, we always assume that the following conditions hold:

(M₁) There existsm₀ >0 such that

M(t)≥m0, ∀t≥0;

(M₂) There existsµ∈(0, 1)such that

Mb(t)≥(1−µ)M(t)t, ∀t ≥0, where Mb(t) =Rt

0 M(s)ds.

Definition 3.3. A functional J is said to satisfy the Palais–Smale condition (or (PS) condition) in a space X, if any sequence {u_n} ⊂ X such that {J(u_n)} is bounded and J⁰(u_n) → 0 as n→∞, has a convergent subsequence.

Lemma 3.4. If M satisfies(M₁)–(M₂), f satisfies(F₀)and the Ambrosetti–Rabinowitz type condition, namely,

(F₁) there exist T₀ >0andθ> ₁^p₋⁺

µ such that

0<θF(x,t)≤ f(x,t)t, ∀x∈ _Ω, |t| ≥T0, then the functional J satisfies the (PS) condition.

Proof. Suppose that{u_n} ⊂X,|J(u_n)| ≤cand J⁰(u_n)→0 inX^∗ asn→∞. We will show that {u_n}is bounded inX. By contradiction, we assume thatku_nk →+_{∞. For}nlarge enough, by

the conditions(F₁),(M₁),(M₂)and Proposition2.2we have c+kunk ≥ J(un)−¹

θJ⁰(un)(un)

=

Z

Ω

1

p(x)|_∆un|^p(x)dx+Mb Z

Ω

1

p(x)|∇un|^p(x)dx

−

Z

ΩF(x,un)dx

− ¹ θ

Z

Ω|_∆un|^p(x)dx−¹ θM

_Z

Ω

1

p(x)|∇u_n|^p(x)dx _Z

Ω|∇u_n|^p(x)dx + ¹

θ Z

Ω f(x,u_n)u_ndx

≥ 1

p⁺ − ¹ θ

_Z

Ω|_∆un|^p(x)dx+ (1−µ)M _Z

Ω

1

p(x)|∇u_n|^p(x)dx _Z

Ω

1

p(x)|∇u_n|^p(x)dx

− ¹ θM

_Z

Ω

1

p(x)|∇u_n|^p(x)dx _Z

Ω|∇u_n|^p(x)dx +

Z

Ω

1

θf(x,un)un−F(x,un)

dx

≥ 1

p⁺ − ¹ θ

_Z

Ω|_∆un|^p(x)dx+m₀

1−µ p⁺ − ¹

θ _Z

Ω|∇u_n|^p(x)dx +

Z

{x∈_Ω:|un|≥T0}

1

θ f(x,u_n)u_n−F(x,u_n)

dx−c₃

≥c₄ku_nk^p−−c₃, wherec₄=min ₁

p⁺ −¹_θ,m₀ ¹_p⁻+^µ −¹_θ > 0 sinceθ> ₁^p₋⁺_µ > p⁺>1.

Dividing by ku_nk^p− in the last inequality and letting n → _∞ we obtain a contradiction.

It follows that the sequence {un} is bounded in X. Without loss of generality, we assume that {u_n}converges weakly to u in X. Then {u_n}converges strongly tou in L^r(x)(_Ω)for all r(x)< p^∗₂(x)_{. Since} J⁰(u_n)→ 0 inX^∗ we deduce that J⁰(u_n)(u_n−u)→ 0 asn → _{∞. We also} haveJ⁰(u)(un−u)→0 asn→_∞because{un}converges weakly touin X. Thus,

nlim→_∞(J⁰(u_n)−J⁰(u))(u_n−u) =0. (3.5) Using(F0)and the Hölder inequality, we have

Z

Ω(f(x,u_n)− f(x,u))(u_n−u)dx

≤

Z

Ω|f(x,u_n)− f(x,u)||u_n−u|dx

≤ C Z

Ω

2+|un|^q(x)−1+|u|^q(x)−1|un−u|dx

≤2C

2+|u_n|^q(x)−1

q⁰(x)+|u|^q(x)−1

q⁰(x)

|u_n−u|_q(x)

→0, q⁰(x) = ^q(x) q(x)−1 whenn→+∞. This implies that

nlim→_∞ Z

Ω(f(x,un)−f(x,u))(un−u)dx=0. (3.6)

Since the sequence {u_n} converges weakly to u ∈ X = W₀^1,p(x)(_Ω)∩W^2,p(x)(_Ω), it is bounded in Xand converges weakly touinW₀^1,p(x)(_Ω), so we deduce that

nlim→∞

M

_Z

Ω|∇u|^p(x)dx

−M _Z

Ω|∇u_n|^p(x)dx _Z

Ω|∇u|^p(x)−2∇u(∇u_n− ∇u)dx= 0.

(3.7) Let us recall the following elementary inequalities (see [6])

|ξ|^s−2ξ− |ζ|^s−2ζ

(ξ−ζ)≥ ¹

2^s|ξ−ζ|^s, s≥2, (3.8)

|ξ|^s−2ξ− |ζ|^s−2ζ

(ξ−ζ) (|ξ|+|ζ|)^2−s≥ (s−1)|ξ−ζ|², 1<s<2 (3.9) for all ξ,ζ ∈_R^N. Put

U_p(x):={x ∈_Ω: p(x)≥2}, V_p(x):={x ∈_Ω: 1< p(x)<2}, (3.10) then, it follows from (3.8) and (3.9) that

Z

U_p(x)

|_∆un−_∆u|^p(x)dx ≤c5

Z

ΩA⁽¹⁾(_∆un,∆u)dx, (3.11) Z

U_p(x)

|∇u_n− ∇u|^p(x)dx≤ c₅ Z

ΩA^(N)(∇u_n,∇u)dx, (3.12) Z

V_p(x)

|_∆un−_∆u|^p(x)dx≤c₆ Z

Ω

A⁽¹⁾(_∆un,∆u)

p(x)

2

C⁽¹⁾(_∆un,∆u)^(2−p(x))

p(x)

2 dx, (3.13) Z

V_p(x)

|∇u_n− ∇u|^p(x)dx≤c₆ Z

Ω

A^(N)(∇u_n,∇u)

p(x)

2

C^(N)(∇u_n,∇u)^(2−p(x))

p(x)

2 dx, (3.14)

where A⁽ⁱ⁾,C⁽ⁱ⁾ :Rⁱ×_Rⁱ →_R,i=1,Nare defined by the following formulas A⁽ⁱ⁾(ξ,ζ):=|ξ|^p(x)−2ξ− |ζ|^p(x)−2ζ

(ξ−ζ), C⁽ⁱ⁾(ξ,ζ):=|ξ|+|ζ|

for allξ,ζ ∈_Rⁱ,i=1,N. Now, from the definition of the functional Jand relations (3.5)–(3.7), we have

0≤

Z

Ω(|_∆u_n|^p(x)−2_∆u_n− |_∆u|^p(x)−2_∆u)(_∆u_n−_∆u)dx +M

_Z

Ω|∇u_n|^p(x)dx _Z

Ω(|∇u_n|^p(x)−2∇u_n− |∇u|^p(x)−2∇u)(∇u_n− ∇u)dx

= (J⁰(u_n)−J⁰(u))(u_n−u) +

Z

Ω(f(x,u_n)− f(x,u))(u_n−u)dx +

M

Z

Ω|∇u|^p(x)dx

−M Z

Ω|∇un|^p(x)dx Z

Ω|∇u|^p(x)−2∇u(∇un− ∇u)dx

→0

whenn→_{∞. By}(M₁), it follows that

nlim→_∞ Z

ΩA⁽¹⁾(_∆un,∆u)dx= lim

n→_∞ Z

ΩA^(N)(∇un,∇u)dx=0. (3.15)

For this reason, we can assume that 0≤ R

ΩA⁽¹⁾(_∆un,∆u)dx < 1. IfR

ΩA⁽¹⁾(_∆un,∆u)dx = 0 then A⁽¹⁾(_∆un,∆u) = 0 since A⁽¹⁾(_∆un,∆u) ≥ 0 in Ω. If 0 < R

ΩA⁽¹⁾(_∆un,∆u)dx < 1, then thanks to the Young inequality

ab≤ ^a

r

r +^b

r⁰

r⁰ , ∀a,b>0, 1 r + ¹

r⁰ =1, r,r⁰ ∈(1,+_∞), with

a=A⁽¹⁾(_∆un,∆u)

p(x)

2 Z

V_p(x)

A⁽¹⁾(_∆un,∆u)dx

!^−p₂^(x)

, b=C⁽¹⁾(_∆un,∆u)^(2−p(x))

p(x)

2 ,

r= ²

p(x)^{, r}

0 = ²

2−p(x)^, we conclude that

Z

V_p(x)

A⁽¹⁾(_∆u_n,∆u)dx

!−¹₂ Z

V_p(x)

A⁽¹⁾(_∆u_n,∆u)

p(x)

2

C⁽¹⁾(_∆u_n,∆u)^(2−p(x))

p(x)

2 dx

≤

Z

V_p(x)

A⁽¹⁾(_∆un,∆u)

p(x)

2 Z

V_p(x)

A⁽¹⁾(_∆un,∆u)dx

!−^p(₂^x)

C⁽¹⁾(_∆un,∆u)^(2−p(x))

p(x)

2 dx

≤

Z

V_p(x)



A⁽¹⁾(_∆un,∆u)

Z

V_p(x)

A⁽¹⁾(_∆un,∆u)_dx

!−¹₂

+_C⁽¹⁾(_∆un,∆u)^p(x)



 dx

≤₁+

Z

Ω

C⁽¹⁾(_∆u_n,∆u)^p(x) dx.

Hence, by relation (3.13), 1

c₆ Z

V_p(x)

|_∆un−_∆u|^p(x)dx≤

Z

V_p(x)

A⁽¹⁾(_∆un,∆u)dx

!¹₂

1+

Z

Ω

C⁽¹⁾(_∆un,∆u)^p(x) dx

. (3.16) We also have

1 c₆

Z

V_p(x)

|∇u_n− ∇u|^p(x)dx≤

Z

V_p(x)

A^(N)(∇u_n,∇u)dx

!¹₂

1+

Z

Ω

C^(N)(∇u_n,∇u)^p(x) dx

. (3.17) By (3.11), (3.13), (3.15) and (3.16), we have

Z

Ω|_∆un−_∆u|^p(x)dx=

Z

U_p(x)

|_∆un−_∆u|^p(x)dx+

Z

V_p(x)

|_∆un−_∆u|^p(x)dx→0 (3.18) whenn→∞. Similarly, from (3.12), (3.14), (3.15) and (3.17) we have

Z

Ω|∇u_n− ∇u|^p(x)dx=

Z

U_p(x)

|∇u_n− ∇u|^p(x)dx+

Z

V_p(x)

|∇u_n− ∇u|^p(x)dx→0. (3.19) Therefore,

ku_n−uk^p+ ≤

Z

Ω

|_∆u_n−_∆u|^p(x)+|∇u_n− ∇u|^p(x) dx→0

whenn→∞. So, the sequence{u_n}converges strongly tou∈ Xand the functionalJsatisfies the (PS) condition inX.

Lemma 3.5. If M satisfies(M₁),(M₂)and f satisfies(F₀),(F₁)and the following condition (_F2) _f(_x,t) =_o(|t|^p+−1)_{for x} ∈_Ωuniformly,

where q⁻> p⁺, then problem(1.1)has a nontrivial weak solution.

Proof. Our idea is to apply the mountain pass theorem [3]. By Lemma 3.4, J satisfies the Palais–Smale condition in X. Since p⁺ < q⁻ ≤ q(x)< p^∗₂(x), the embedding X,→ L^p+(_Ω)_is continuous and compact and then there existsc₇>0 such that

|u|_p⁺ ≤c₇kuk, ∀u∈X. (3.20) Let e > 0 be small enough such that ec₇^p+ < _2p¹+ min{1,m₀}. By the assumptions (F₀) and (F₂), there existsc_e>0 depending onesuch that

|F(x,t)| ≤e|t|^p++c_e|t|^q(x), ∀(x,t)∈_Ω×_R. (3.21) Hence, for all u∈Xwithkuk<1, we have

J(u)≥

Z

Ω

1

p(x)|_∆u|^p(x)dx+Mb _Z

Ω

1

p(x)|∇u|^p(x)dx

−

Z

ΩF(x,u)dx

≥ ¹ p⁺

Z

Ω|_∆u|^p(x)dx+ ^m0 p⁺

Z

Ω|∇u|^p(x)dx−e Z

Ω|u|^p+dx−c_e Z

Ω|u|^q(x)dx

≥ ^min{1,m₀}

p⁺ kuk^p+−ec₇^p+kuk^p+−c_ekuk^q−

≥ ^min{1,m₀}

2p⁺ kuk^p+−cekuk^q−,

where c_e is a positive constant. Sinceq⁻ > p⁺, we conclude that there exist α> 0 andρ >0 such that J(u)≥α>_{0 for all} u∈Xwith kuk= _ρ.

On the other hand, from(F₁)it follows that

F(x,t)≥c₈|t|^θ−c₉, ∀x ∈_Ω, t∈_R. (3.22) From(M₂)we can easily obtain that

Mb(t)≤ ^Mb(t0) t

1−1µ

0

t^1−1µ =c₁₀t^1−1µ, ∀t>t₀, (3.23)

wheret0 is an arbitrary positive constant. Hence, forw∈X\{0}andt >1, we have J(tw) =

Z

Ω

1

p(x)|t∆w|^p(x)dx+Mb _Z

Ω

1

p(x)|t∇w|^p(x)dx

−

Z

ΩF(x,tw)dx

≤ ^t

p⁺

p⁻ Z

Ω|_∆w|^p(x)dx+c₁₀t^p

+ 1−µ

_Z

Ω|∇w|^p(x)dx ₁₋¹

µ −c₈t^θ Z

Ω|w|^θdx−c₉

→ −_∞ ast →+_∞,

due to θ > ₁^p₋⁺_µ > p⁺. Since J(0) = 0, we conclude that J satisfies all assumptions of the mountain pass theorem [3]. So, J admits at least one nontrivial critical point and problem (1.1) has a nontrivial weak solution.