Existence Results for the p(x)-Laplacian with Nonlinear Boundary Condition

Abstract

By using the variational method, under appropriate assumptions on the perturbation terms 𝑓(𝑥,𝑢),𝑔(𝑥,𝑢) such that the associated functional satisfies the global minimizer condition and the fountain theorem, respectively, the existence and multiple results for the 𝑝(𝑥)-Laplacian with nonlinear boundary condition in bounded domain Ω were studied. The discussion is based on variable exponent Lebesgue and Sobolev spaces.

1. Introduction

In recent years, increasing attention has been paid to the study of differential and partial differential equations involving variable exponent conditions. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, or calculus of variations. For more information on modeling physical phenomena by equations involving 𝑝(𝑥)-growth condition we refer to [1–3]. The appearance of such physical models was facilitated by the development of variable exponent Lebesgue and Sobolev spaces, 𝐿^𝑝(𝑥) and 𝑊^1,𝑝(𝑥), where 𝑝(𝑥) is a real-valued function. Variable exponent Lebesgue spaces appeared for the first time in the literature as early as 1931 in an article by Orlicz [4]. The spaces 𝐿^𝑝(𝑥) are special cases of Orlicz spaces 𝐿^𝜑 originated by Nakano [5] and developed by Musielak and Orlicz [6, 7], where 𝑓∈𝐿^𝜑 if and only if ∫𝜑(𝑥,|𝑓(𝑥)|)𝑑𝑥<∞ for a suitable 𝜑. Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers. In that context we refer to the studies of Tsenov [8], Sharapudinov [9], and Zhikov [10, 11].

In this paper, we consider the following nonlinear elliptic boundary value problem: −di𝑎v(𝑥𝑎)||(∇𝑥)𝑢||||∇𝑢||^{𝑝(𝑥)−2}_{𝑝(𝑥)−2}∇𝑢+𝑏(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢=𝜆𝑓(𝑥,𝑢),𝑥∈Ω,𝜕𝑢𝜕𝜈=𝑐(𝑥)|𝑢|^{𝑞(𝑥)−2}𝑢+𝜇𝑔(𝑥,𝑢),𝑥∈𝜕Ω,(1.1) where Ω⊂ℝ^𝑛 is a bounded domain with Lipschitz boundary 𝜕Ω,𝜕/𝜕𝜈 is outer unit normal derivative, 𝑝(𝑥)∈𝐶(Ω),𝑞(𝑥)∈𝐶(𝜕Ω),𝑝(𝑥),𝑞(𝑥)>1, and 𝑝(𝑥)≠𝑞(𝑦) for any 𝑥∈Ω,𝑦∈𝜕Ω;𝜆,𝜇∈ℝ;𝑓∶Ω×ℝ→ℝ, and 𝑔∶𝜕Ω×ℝ→ℝ are Carathédory functions. Throughout this paper, we assume that 𝑎(𝑥),𝑏(𝑥), and 𝑐(𝑥) satisfy 0<𝑎₁≤𝑎(𝑥)≤𝑎₂,0<𝑏₁≤𝑏(𝑥)≤𝑏₂, and 0≤𝑐₁≤𝑐(𝑥)≤𝑐₂.

The operator −Δ_𝑝(𝑥)𝑢∶=−div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) is called 𝑝(𝑥)-Laplacian, which is a natural extension of the 𝑝-Laplace operator, with 𝑝 being a positive constant. However, such generalizations are not trivial since the 𝑝(𝑥)-Laplace operator possesses a more complicated structure than the 𝑝-Laplace operator, for example, it is inhomogeneous. For related results involving the Laplace operator, see [12, 13].

In the past decade, many people have studied the nonlinear boundary value problems involving 𝑝-Laplacian. For example, if 𝜆=𝜇=1,𝑎(𝑥)=𝑏(𝑥)=𝑐(𝑥)≡1,𝑝(𝑥)≡𝑝, and 𝑞(𝑥)≡𝑞 (a constant), then problem (1.1) becomes −div||∇𝑢||||∇𝑢_𝑝−2||^𝑝−2∇𝑢+|𝑢|^𝑝−2𝑢=𝑓(𝑥,𝑢),𝑥∈Ω,𝜕𝑢𝜕𝜈=|𝑢|^𝑞−2𝑢+𝑔(𝑥,𝑢),𝑥∈𝜕Ω.(1.2) Bonder and Rossi [14] considered the existence of nontrivial solutions of problem (1.2) when 𝑓(𝑥,𝑢)≡0 and discussed different cases when 𝑔(𝑥,𝑢) is subcritical, critical, and supercritical with respect to 𝑢. We also mention that Martínez and Rossi [15] studied the existence of solutions when 𝑝=𝑞 and the perturbation terms 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢) satisfy the Landesman-Lazer-type conditions. Recently, J.-H. Zhao and P.-H. Zhao [16] studied the nonlinear boundary value problem, assumed that 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢) satisfy the Ambrosetti-Rabinowitz-type condition, and got the multiple results.

If 𝜆=𝜇=1,𝑝(𝑥)≡𝑝, and 𝑞(𝑥)≡𝑞 (a constant), then problem (1.1) becomes −div𝑎(𝑥𝑎)||(∇𝑥)𝑢||||∇𝑢_𝑝−2||^𝑝−2∇𝑢+𝑏(𝑥)|𝑢|^𝑝−2𝑢=𝑓(𝑥,𝑢),𝑥∈Ω,𝜕𝑢𝜕𝜈=𝑐(𝑥)|𝑢|^𝑞−2𝑢+𝑔(𝑥,𝑢),𝑥∈𝜕Ω.(1.3) There are also many people who studied the 𝑝-Laplacian nonlinear boundary value problems involving (1.3). For example, Cîrstea and Rǎdulescu [17] used the weighted Sobolev space to discuss the existence and nonexistence results and assumed that 𝑓(𝑥,𝑢) is a special case in the problem (1.3), where Ω is an unbounded domain. Pflüger [18], by using the same technique, considered the existence and multiplicity of solutions when 𝑏(𝑥)≡0. The author showed the existence result when 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢) are superlinear and satisfy the Ambrosetti-Rabinowitz-type condition and got the multiplicity of solutions when one of 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢) is sublinear and the other one is superlinear.

More recently, the study on the nonlinear boundary value problems with variable exponent has received considerable attention. For example, Deng [19] studied the eigenvalue of 𝑝(𝑥)-Laplacian Steklov problem, and discussed the properties of the eigenvalue sequence under different conditions. Fan [20] discussed the boundary trace embedding theorems for variable exponent Sobolev spaces and some applications. Yao [21] constrained the two nonlinear perturbation terms 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢) in appropriate conditions and got a number of results for the existence and multiplicity of solutions. Motivated by Yao and problem (1.3), we consider the more general form of the variable exponent boundary value problem (1.1). Under appropriate assumptions on the perturbation terms 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢), by using the global minimizer method and fountain theorem, respectively, the existence and multiplicity of solutions of (1.1) were obtained. These results extend some of the results in [21] and the classical results for the 𝑝-Laplacian in [14, 16, 22–24].

2. Preliminaries

In order to discuss problem (1.1), we need some results for the spaces 𝑊^1,𝑝(𝑥)(Ω), which we call variable exponent Sobolev spaces. We state some basic properties of the spaces 𝑊^1,𝑝(𝑥)(Ω), which will be used later (for more details, see [25, 26]). Let Ω be a bounded domain of ℝ^𝑛, and denote 𝐶₊Ω=𝑝(𝑥)∣𝑝(𝑥)∈𝐶Ω;𝑝(𝑥)>1,∀𝑥∈Ω.(2.1) For 𝑝(𝑥)∈𝐶₊(Ω) write 𝑝⁺=max_𝑥∈Ω𝑝(𝑥),𝑝⁻=min_𝑥∈Ω𝑝(𝑥).(2.2) We can also denote 𝐶₊(𝜕Ω) and 𝑞⁺,𝑞⁻ for any 𝑞(𝑥)∈𝐶(𝜕Ω), and define 𝐿^𝑝(𝑥)𝐿(Ω)=𝑢∣𝑢isameasurablereal-valuedfunction,_Ω||𝑢(𝑥)||^𝑝(𝑥)𝑑𝑥<∞_𝑝(𝑥),(𝜕Ω)=𝑢∣𝑢∶𝜕Ω⟶ℝisameasurablereal-valuedfunction,_𝜕Ω||𝑢(𝑥)||^𝑝(𝑥)𝑑𝜎<∞,(2.3) with norms on 𝐿^𝑝(𝑥)(Ω) and 𝐿^𝑝(𝑥)(𝜕Ω) defined by|𝑢|_𝐿^𝑝(𝑥)_(Ω)=|𝑢|_𝑝(𝑥)=inf𝜆>0∶_Ω|||𝑢(𝑥)𝜆|||^𝑝(𝑥)|𝑢|_{𝐿𝑝(𝑥)(𝜕Ω)}𝑑𝑥≤1,=inf𝜏>0∶_𝜕Ω|||𝑢(𝑥)𝜏|||^𝑝(𝑥)𝑑𝜎≤1,(2.4) where 𝑑𝜎 is the surface measure on 𝜕Ω. Then, (𝐿^𝑝(𝑥)(Ω),|⋅|_𝑝(𝑥)) and (𝐿^𝑝(𝑥)(𝜕Ω),|⋅|_𝐿^𝑝(𝑥)_(𝜕Ω)) become Banach spaces, which we call variable exponent Lebesgue spaces. Let us define the space 𝑊^1,𝑝(𝑥)(Ω)=𝑢∈𝐿^𝑝(𝑥)(Ω)∶||∇𝑢||∈𝐿^𝑝(𝑥)(Ω),(2.5) equipped with the norm ‖𝑢‖=inf𝜆>0∶_Ω|||∇𝑢𝜆(𝑥)|||^𝑝(𝑥)+|||𝑢(𝑥)𝜆|||^𝑝(𝑥)𝑑𝑥≤1.(2.6) For 𝑢∈𝑊^1,𝑝(𝑥)(Ω), if we define ‖𝑢‖^=inf𝜆>0∶_Ω𝑎(𝑥)|||∇𝑢𝜆(𝑥)|||^𝑝(𝑥)+𝑏(𝑥)|||𝑢(𝑥)𝜆|||^𝑝(𝑥)𝑑𝑥≤1,(2.7) then, from the assumptions of 𝑎(𝑥) and 𝑏(𝑥), it is easy to check that ‖𝑢‖^ is an equivalent norm on 𝑊^1,𝑝(𝑥)(Ω). For simplicity, we denote Γ(𝑢)=_Ω𝑎(𝑥)||∇𝑢||^𝑝(𝑥)+|𝑢|^𝑝(𝑥)𝑑𝑥.(2.8)

Hence, we have (see [27])(i)if Γ(𝑢)≥1, then 𝜉₁‖𝑢‖^𝑝−≤Γ(𝑢)≤𝜉₂‖𝑢‖^𝑝+,(ii)if Γ(𝑢)≤1, then 𝜁₁‖𝑢‖^𝑝+≤Γ(𝑢)≤𝜁₂‖𝑢‖^𝑝−,where 𝜉₁,𝜉₂ and 𝜁₁,𝜁₂ are positive constants independent of 𝑢.

Denote by 𝑊^{01,𝑝(𝑥)}(Ω) the closure of 𝐶^∞0(Ω) in 𝑊^1,𝑝(𝑥)(Ω).

Proposition 2.1 (see [21, 28]). (1) The space (𝐿^𝑝(𝑥)(Ω), |⋅|_𝑝(𝑥)) is a separable, uniformly convex Banach space, and its conjugate space is 𝐿^𝑞(𝑥)(Ω), where 1/𝑞(𝑥)+1/𝑝(𝑥)=1. For any 𝑢∈𝐿^𝑝(𝑥)(Ω) and 𝑣∈𝐿^𝑞(𝑥)(Ω), one has ||||_Ω𝑢𝑣d𝑥||||≤1𝑝⁻+1𝑞⁻|𝑢|^𝑝(𝑥)|𝑣|_𝑞(𝑥).(2.9)

(2) If 𝑝₁,𝑝₂∈𝐶₊(Ω), 𝑝₁(𝑥)≤𝑝₂(𝑥), for any 𝑥∈Ω, then 𝐿^𝑝2(𝑥)(Ω)↪𝐿^𝑝1(𝑥)(Ω) and the imbedding is continuous.

Proposition 2.2 (see [20, 21, 28]). (1)   𝑊^1,𝑝(𝑥)(Ω),𝑊^{01,𝑝(𝑥)}(Ω) are separable reflexive Banach spaces.
(2) If 𝑞(𝑥)∈𝐶₊(Ω) and 𝑞(𝑥)<𝑝^∗(𝑥) for any 𝑥∈Ω, then the embedding from 𝑊^1,𝑝(𝑥)(Ω) into 𝐿^𝑞(𝑥)(Ω) is compact and continuous, where 𝑝^∗(𝑥)=⎧⎪⎨⎪⎩∞𝑛𝑛−,𝑝𝑝(𝑥()𝑥),iiff𝑝𝑝((𝑥𝑥))<≥𝑛𝑛,.(2.10)
(3) If 𝑞(𝑥)∈𝐶₊(𝜕Ω) and 𝑞(𝑥)<𝑝_∗(𝑥) for any 𝑥∈𝜕Ω, then the trace imbedding from 𝑊^1,𝑝(𝑥)(Ω) into 𝐿^𝑞(𝑥)(𝜕Ω) is compact and continuous, where 𝑝_∗(𝑥)=⎧⎪⎨⎪⎩∞(𝑛𝑛,−−1𝑝)(𝑝𝑥()𝑥),iiff𝑝𝑝((𝑥𝑥))<≥𝑛𝑛,.(2.11)
(4) (Poincaré inequality) There is a constant 𝐶>0, such that |𝑢|_𝑝(𝑥)≤𝐶||∇𝑢||_𝑝(𝑥)∀𝑢∈𝑊^{01,𝑝(𝑥)}(Ω).(2.12)

Proposition 2.3 (see [21, 28, 29]). If 𝑓∶Ω×ℝ→ℝ is a Carathéodory function and satisfies ||𝑓(𝑥,𝑠)||≤𝑎(𝑥)+𝑏|𝑠|^{𝑝1(𝑥)/𝑝2(𝑥)},forany𝑥∈Ω,𝑠∈ℝ,(2.13) where 𝑝₁(𝑥), 𝑝₂(𝑥)∈𝐶₊(Ω), 𝑎(𝑥)∈𝐿^𝑝2(𝑥)(Ω), 𝑎(𝑥)≥0, and 𝑏≥0 is a constant, then the Nemytsky operator from 𝐿^𝑝1(𝑥)(Ω) to 𝐿^𝑝2(𝑥)(Ω) defined by (𝑁_𝑓(𝑢))(𝑥)=𝑓(𝑥,𝑢(𝑥)) is a continuous and bounded operator.

Proposition 2.4 (see [21, 28, 30]). Denote 𝜌(𝑢)=_Ω|𝑢|^𝑝(𝑥)d𝑥,∀𝑢∈𝐿^𝑝(𝑥)(Ω).(2.14) Then,
(1)|𝑢|_𝑝(𝑥)<1(=1;>1) if and only if 𝜌(𝑢)<1(=1;>1),
(2)  |𝑢|_𝑝(𝑥)>1 implies |𝑢|^𝑝−_𝑝(𝑥)≤𝜌(𝑢)≤|𝑢|^𝑝+_𝑝(𝑥) and |𝑢|_𝑝(𝑥)<1 implies |𝑢|^𝑝−_𝑝(𝑥)≥𝜌(𝑢)≥|𝑢|^𝑝+_𝑝(𝑥),
(3)|𝑢|_𝑝(𝑥)→0 if and only if 𝜌(𝑢)→0 and |𝑢(𝑥)|_𝑝(𝑥)→∞ if and only if 𝜌(𝑢)→∞.

Proposition 2.5 (see [19]). Denote 𝜌(𝑢)=_𝜕Ω|𝑢|^𝑝(𝑥)d𝜎,∀𝑢∈𝐿^𝑝(𝑥)(𝜕Ω).(2.15) Then, (1)|𝑢|_𝐿^𝑝(𝑥)_(𝜕Ω)>1 implies |𝑢|^𝑝−_{𝐿𝑝(𝑥)(𝜕Ω)}≤𝜌(𝑢)≤|𝑢|^𝑝+_{𝐿𝑝(𝑥)(𝜕Ω)},(2)|𝑢|_𝐿^𝑝(𝑥)_(𝜕Ω)<1 implies |𝑢|^𝑝−_{𝐿𝑝(𝑥)(𝜕Ω)}≥𝜌(𝑢)≥|𝑢|^𝑝+_{𝐿𝑝(𝑥)(𝜕Ω)}.

3. Assumptions and Statement of Main Results

In the following, let 𝑋 denote the generalized Sobolev space 𝑊^1,𝑝(𝑥)(Ω),  𝑋^∗ denote the dual space of 𝑊^1,𝑝(𝑥)(Ω), ⟨⋅⟩ denote the dual pair, and let → represent strong convergence, ⇀ represent weak convergence, 𝐶, 𝐶_𝑖 represent the generic positive constants.

Now we state the assumptions on perturbation terms 𝑓(𝑥,𝑢) and 𝑔(𝑥,𝑢) for problem (1.1) as follows: (𝑓₀) 𝑓∶Ω×ℝ→ℝ satisfies Carathéodory condition and there exist two constants 𝑐₁≥0,𝑐₂>0 such that ||𝑓(𝑥,𝑢)||≤𝑐₁+𝑐₂|𝑢|^{𝛼(𝑥)−1},∀(𝑥,𝑢)∈Ω×ℝ,(3.1) where 𝛼(𝑥)∈𝐶₊(Ω) and 𝛼(𝑥)<𝑝^∗(𝑥) for any 𝑥∈Ω.(𝑓₁) There exist 𝑀₁>0,𝜃₁>𝑝⁺ such that 0<𝜃₁𝐹(𝑥,𝑢)≤𝑓(𝑥,𝑢)𝑢,|𝑢|≥𝑀₁,∀𝑥∈Ω.(3.2)(𝑓₂)𝑓(𝑥,−𝑢)=−𝑓(𝑥,𝑢),forall𝑥∈Ω,𝑢∈ℝ.(𝑔₀)𝑔∶𝜕Ω×ℝ→ℝ satisfies Carathéodory condition and there exist two constants 𝑐^1≥0,𝑐^2>0 such that ||𝑔(𝑥,𝑢)||≤𝑐^1+𝑐^2|𝑢|^{𝛽(𝑥)−1},∀(𝑥,𝑢)∈𝜕Ω×ℝ,(3.3) where 𝛽(𝑥)∈𝐶₊(𝜕Ω) and 𝛽(𝑥)<𝑝_∗(𝑥) for any 𝑥∈𝜕Ω.(𝑔₁) There exist 𝑀₂>0,𝜃₂>𝑝⁺ such that 0<𝜃₂𝐺(𝑥,𝑢)≤𝑔(𝑥,𝑢)𝑢,|𝑢|≥𝑀₂,∀𝑥∈𝜕Ω.(3.4)(𝑔₂)𝑔(𝑥,−𝑢)=−𝑔(𝑥,𝑢),forall𝑥∈𝜕Ω,𝑢∈ℝ.

The functional associated with problem (1.1) is 𝜑(𝑢)=_Ω𝑎(𝑥)||∇𝑢||^𝑝(𝑥)+𝑏(𝑥)|𝑢|^𝑝(𝑥)𝑝(𝑥)d𝑥−𝜆_Ω−𝐹(𝑥,𝑢)𝑑𝑥_𝜕Ω𝑐(𝑥)𝑞(𝑥)|𝑢|^𝑞(𝑥)d𝜎−𝜇_𝜕Ω𝐺(𝑥,𝑢)𝑑𝜎,(3.5) where 𝐹(𝑥,𝑢) and 𝐺(𝑥,𝑢) are denoted by 𝐹(𝑥,𝑢)=^𝑢0𝑓(𝑥,𝑠)d𝑠,𝐺(𝑥,𝑢)=^𝑢0𝑔(𝑥,𝑠)𝑑𝑠.(3.6) By Propositions 3.1 and 3.2, and assumptions (𝑓₀), (𝑔₀), it is easy to see that the functional 𝜑∈𝐶¹(𝑋,ℝ); moreover, 𝜑 is even if (𝑓₂) and (𝑔₃) hold. Then, 𝜑^(𝑢),𝑣=−_Ω𝑎(𝑥)||∇𝑢||^{𝑝(𝑥)−2}∇𝑢∇𝑣+𝑏(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢𝑣𝑑𝑥−𝜆_Ω𝑓(𝑥,𝑢)𝑣𝑑𝑥_𝜕Ω𝑐(𝑥)|𝑢|^{𝑞(𝑥)−2}𝑢𝑣𝑑𝜎−𝜇_𝜕Ω𝑔(𝑥,𝑢)𝑣𝑑𝜎,(3.7) so the weak solution of (1.1) corresponds to the critical point of the functional 𝜑.

Before giving our main results, we first give several propositions that will be used later.

Proposition 3.1 (see [31]). If one denotes 𝐼(𝑢)=_Ω𝑎(𝑥)||∇𝑢||^𝑝(𝑥)+𝑏(𝑥)|𝑢|^𝑝(𝑥)𝑝(𝑥)d𝑥,∀𝑢∈𝑋,(3.8) then 𝐼∈𝐶¹(𝑋,ℝ) and the derivative operator of 𝐼, denoted by 𝐼′, is 𝐼^(𝑢),𝑣=_Ω𝑎(𝑥)||∇𝑢||^{𝑝(𝑥)−2}∇𝑢∇𝑣+𝑏(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢𝑣d𝑥,∀𝑢,𝑣∈𝑋,(3.9) and one has:(i)𝐼′∶𝑋→𝑋^∗ is a continuous, bounded, and strictly monotone operator,(ii)𝐼′ is a mapping of (S+) type, that is, if 𝑢_𝑛⇀𝑢 in 𝑋 and limsup_𝑛→∞⟨𝐼′(𝑢_𝑛)−𝐼′(𝑢),𝑢_𝑛−𝑢⟩≤0, then 𝑢_𝑛→𝑢 in 𝑋,(iii)𝐼′∶𝑋→𝑋^∗ is a homeomorphism.

Proposition 3.2 (see [19]). If one denotes 𝐽(𝑢)=_𝜕Ω𝑐(𝑥)𝑞(𝑥)|𝑢|^𝑞(𝑥)d𝜎,∀𝑢∈𝑋,(3.10) where 𝑞(𝑥)∈𝐶₊(𝜕Ω) and 𝑞(𝑥)<𝑝_∗(𝑥) for any 𝑥∈𝜕Ω, then 𝐽∈𝐶¹(𝑋,ℝ) and the derivative operator 𝐽′ of 𝐽 is 𝐽^(𝑢),𝑣=_𝜕Ω𝑐(𝑥)|𝑢|^{𝑞(𝑥)−2}𝑢𝑣d𝜎,∀𝑢,𝑣∈𝑋,(3.11) and one has that 𝐽∶𝑋→ℝ and 𝐽′∶𝑋→𝑋^∗ are sequentially weakly-strongly continuous, namely, 𝑢_𝑛⇀𝑢 in 𝑋 implies 𝐽′(𝑢_𝑛)→𝐽′(𝑢).

Let 𝑋 be a reflexive and separable Banach space. There exist 𝑒_𝑖∈𝑋 and 𝑒^∗𝑗∈𝑋^∗ such that 𝑋=span𝑒_𝑖∶𝑖=1,2𝑒,…,𝑋^∗=span𝑒^∗𝑗∶𝑗=1,2,…_𝑖,,𝑒^∗𝑗=10,,𝑖𝑖≠=𝑗𝑗.,(3.12) For 𝑘=1,2,…, denote 𝑋_𝑘=span𝑒_𝑘,𝑌_𝑘=^𝑘_𝑖=1𝑋_𝑖,𝑍_𝑘=_𝑖≥𝑘𝑋_𝑖.(3.13)

One important aspect of applying the standard methods of variational theory is to show that the functional 𝜑 satisfies the 𝑃𝑎𝑙𝑎𝑖𝑠-𝑆𝑚𝑎𝑙𝑒 condition, which is introduced by the following definition.

Definition 3.3. Let 𝜑∈𝐶¹(𝑋,ℝ) and 𝑐∈ℝ. Then, functional 𝜑 satisfies the (PS)_𝑐 condition if any sequence {𝑢_𝑛}⊂𝑋 such that 𝜑𝑢_𝑛⟶𝑐,𝜑^𝑢_𝑛⟶0in𝑋^∗,as𝑛⟶∞(3.14) contains a subsequence converging to a critical point of 𝜑.
In what follows we write the (PS)_𝑐 condition simply as the (PS) condition if it holds for every level 𝑐∈ℝ for the 𝑃𝑎𝑙𝑎𝑖𝑠-𝑆𝑚𝑎𝑙𝑒 condition at level 𝑐.

Proposition 3.4 (Fountain theorem, see [23, 32]). Assume that(A1)𝑋 is a Banach space, 𝜑∈𝐶¹(𝑋,ℝ) is an even functional, the subspaces 𝑋_𝑘,𝑌_𝑘 and 𝑍_𝑘 are defined by (3.13).Suppose that, for every 𝑘∈𝐍, there exist 𝜌_𝑘>𝛾_𝑘>0 such that(A2)inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝛾𝑘}𝜑(𝑢)→∞ as 𝑘→∞,(A3)max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝜑(𝑢)≤0,(A4)𝜑 satisfies (PS)_𝑐 condition for every 𝑐>0.

Then, 𝜑 has a sequence of critical values tending to +∞.

Proposition 3.5 (see [21]). Suppose that hypotheses 𝛼(𝑥)∈𝐶₊(Ω),𝛼(𝑥)<𝑝^∗(𝑥),forall𝑥∈Ω, and if 𝑞(𝑥)∈𝐶₊(𝜕Ω),𝑞(𝑥)<𝑝_∗(𝑥),forall𝑥∈𝜕Ω, denote 𝛼𝑞_𝑘𝑘=sup|𝑢|_𝐿^𝛼(𝑥)_(Ω)∶‖𝑢‖=1,𝑢∈𝑍_𝑘;=sup|𝑢|_𝐿^𝑞(𝑥)_(𝜕Ω)∶‖𝑢‖=1,𝑢∈𝑍_𝑘,(3.15) then lim_𝑘→∞𝛼_𝑘=0,lim_𝑘→∞𝑞_𝑘=0.
Let us introduce the following lemma that will be useful in the proof of our main result.

Lemma 3.6. Let 𝜆,𝜇≥0,𝑞⁻>𝜃₁,𝜃₂, and assume that (𝑓₀),(𝑓₁),(𝑔₀),and(𝑔₁) are satisfied, then 𝜑 satisfies (PS) condition.

Proof. By Propositions 2.2 and 2.3, we know that if we denote Φ(𝑢)=𝜆_Ω𝐹(𝑥,𝑢)𝑑𝑥+𝜇_𝜕Ω𝐺(𝑥,𝑢)𝑑𝜎,(3.16) then Φ is weakly continuous and its derivative operator, denoted by Φ′, is compact. By Propositions 3.1 and 3.2, we deduce that 𝜑′=𝐼′−𝐽′−Φ′ is also of (S+) type. To verify that 𝜑 satisfies (PS) condition on 𝑋, it is enough to verify that any (PS) sequence is bounded. Suppose that {𝑢_𝑛}⊂𝑋 such that 𝜑𝑢_𝑛⟶𝑐,𝜑^𝑢_𝑛⟶0,in𝑋^∗,as𝑛⟶∞.(3.17) Then, for 𝑛 large enough, we can find 𝑀₃>0 such that ||𝜑𝑢_𝑛||≤𝑀₃.(3.18) Since 𝜑′(𝑢_𝑛)→0, we have ⟨𝜑′(𝑢_𝑛),𝑢_𝑛⟩→0. In particular, {⟨𝜑′(𝑢_𝑛),𝑢_𝑛⟩} is bounded. Thus, there exists 𝑀₄>0 such that ||𝜑^𝑢_𝑛,𝑢_𝑛||≤𝑀₄.(3.19) We claim that the sequence {𝑢_𝑛} is bounded. If it is not true, by passing a subsequence if necessary, we may assume that ‖𝑢_𝑛‖→+∞. Without loss of generality, we assume that ‖𝑢_𝑛‖≥1 appropriately large such that 𝜉₁‖𝑢‖^𝑝−<𝜁₁‖𝑢‖^𝑝+ for any 𝑥∈Ω. From (3.18) and (3.19) and letting 𝜃=min{𝜃₁,𝜃₂}, then 𝜃<𝑞⁻, we have 𝑀₃≥≥𝜑1𝑢_𝑛=𝐼𝑢_𝑛−𝐽𝑢_𝑛−Φ𝑢_𝑛𝑝⁺Γ𝑢_𝑛−1𝑞⁻_𝜕Ω≥1𝑐(𝑥)||𝑢_𝑛||^𝑞(𝑥)𝑑𝜎−Φ𝑢_𝑛,𝑝⁺Γ𝑢_𝑛−1𝜃_𝜕Ω𝑀𝑐(𝑥)||𝑢_𝑛||^𝑞(𝑥)𝑑𝜎−Φ𝑢_𝑛,(3.20)₄≥−𝜑^𝑢_𝑛,𝑢_𝑛=−Γ𝑢_𝑛+_𝜕Ω𝑐(𝑥)||𝑢_𝑛||^𝑞(𝑥)𝑑𝜎+Φ^𝑢_𝑛,𝑢_𝑛.(3.21) By virtue of assumptions (𝑓₁) and (𝑔₁) and combining (3.20) and (3.21), we have 𝜃𝑀₃+𝑀₄≥𝜃𝑝⁺≥𝜃−1Γ𝑢_𝑛−𝜃Φ𝑢_𝑛+Φ^𝑢_𝑛,𝑢_𝑛𝑝⁺+𝜇_𝜕Ω−1𝜉₁‖‖𝑢_𝑛‖‖^𝑝−+𝜆_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−𝜃𝐹𝑥,𝑢_𝑛𝑑𝑥≥𝜃𝑔𝑥,𝑢_𝑛𝑢_𝑛−𝜃𝐺𝑥,𝑢_𝑛𝑑𝜎𝑝⁺−1𝜉₁‖‖𝑢_𝑛‖‖^𝑝−−𝐶.(3.22) Note that 𝜃=min{𝜃₁,𝜃₂}>𝑝⁺, let 𝑛→∞ we obtian a contradiction. It follows that the sequence {𝑢_𝑛} is bounded in 𝑋. Therefore, 𝜑 satisfies (PS) condition.

Under appropriate assumptions on the perturbation terms 𝑓(𝑥,𝑢),𝑔(𝑥,𝑢), a sequence of weak solutions with energy values tending to +∞ was obtained. The main result of the paper reads as follows.

Theorem 3.7. Let 𝛼⁻,𝛽⁻>𝑝⁺,𝑞⁻>𝜃₁,𝜃₂, and 𝜆,𝜇≥0, and assumed that (𝑓₀)−(𝑓₂),(𝑔₀)−(𝑔₂) are satisfied; then 𝜑 has a sequence of critical points {±𝑢_𝑛} such that 𝜑(±𝑢_𝑛)→∞ as 𝑛→∞.

Proof. We will prove that 𝜑 satisfies the conditions of Proposition 3.4. Obviously, because of the assumptions of (𝑓₂) and (𝑔₂), 𝜑 is an even functional and satisfies (PS) condition (see Lemma 3.6). We will prove that if 𝑘 is large enough, then there exist 𝜌_𝑘>𝛾_𝑘>0 such that (A2) and (A3) hold. By virtue of (𝑓₀), (𝑔₀), there exist two positive constants 𝐶₁,𝐶₂ such that ||𝐹(𝑥,𝑢)||≤𝐶₁1+|𝑢|^𝛼(𝑥),(𝑥,𝑢)∈Ω×ℝ;||𝐺(𝑥,𝑢)||≤𝐶₂1+|𝑢|^𝛽(𝑥),(𝑥,𝑢)∈𝜕Ω×ℝ.(3.23) Letting 𝑢∈𝑍_𝑘 with ‖𝑢‖>1 appropriately large such that 𝜉₁‖𝑢‖^𝑝−<𝜁₁‖𝑢‖^𝑝+, we have 𝜑(𝑢)=≥𝐼1(𝑢)−𝐽(𝑢)−Φ(𝑢)𝑝⁺Γ(𝑢)−𝑐²𝑞⁻_𝜕Ω≥1|𝑢|^𝑞(𝑥)𝑑𝜎−𝜆_Ω𝐶₁1+|𝑢|^𝛼(𝑥)𝑑𝑥−𝜇_𝜕Ω𝐶₂1+|𝑢|^𝛽(𝑥)𝑑𝜎𝑝⁺min𝜉₁‖𝑢‖^𝑝−,𝜁₁‖𝑢‖^𝑝+−𝑐²𝑞⁻−𝜆𝐶₁max|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}max|𝑢|^𝛼+_{𝐿𝛼(𝑥)(Ω)},|𝑢|^𝛼−_{𝐿𝛼(𝑥)(Ω)}−𝜇𝐶²max|𝑢|^𝛽+_{𝐿𝛽(𝑥)(𝜕Ω)},|𝑢|^𝛽−_{𝐿𝛽(𝑥)(𝜕Ω)}−𝐶³≥𝜉¹𝑝⁺‖𝑢‖^𝑝−−𝐶(𝑞⁻,𝜆,𝜇)|m𝑢|ax|_{𝛼−𝐿𝛼(𝑥)(Ω)}𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝛼+_{𝐿𝛼(𝑥)(Ω)},,|𝑢|^𝛽+_{𝐿𝛽(𝑥)(𝜕Ω)},|𝑢|^𝛽−_{𝐿𝛽(𝑥)(𝜕Ω)}−𝐶³.(3.24) If max{|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝛼+_{𝐿𝛼(𝑥)(Ω)},|𝑢|^𝛼−_{𝐿𝛼(𝑥)(Ω)},|𝑢|^𝛽+_{𝐿𝛽(𝑥)(𝜕Ω)},|𝑢|^𝛽−_{𝐿𝛽(𝑥)(𝜕Ω)}}=|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)}, then by Proposition 3.5, we have 𝜑𝑢_𝑛≥𝜉¹𝑝⁺‖𝑢‖^𝑝−−𝐶(𝑞⁻,𝜆,𝜇)|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)}−𝐶₃≥𝜉¹𝑝⁺‖𝑢‖^𝑝−−𝐶(𝑞⁻,𝜆,𝜇)𝑞^𝑞+_𝑘‖𝑢‖^𝑞+−𝐶₃.(3.25) Choose 𝛾_𝑘=(𝑞⁺𝐶(𝑞⁻,𝜆,𝜇)(𝑞_𝑘⁽𝑞⁺))/𝜉_1)^{1/(𝑝−−𝑞+)}. For 𝑢∈𝑍_𝑘 with ‖𝑢‖=𝛾_𝑘, we have 𝜑(𝑢)≥𝜉₁𝑝1⁺−1𝑞⁺𝛾^𝑝−𝑘−𝐶₃.(3.26) Since 𝑞_𝑘→0 as 𝑘→∞ and 1<𝑝⁻≤𝑝⁺<𝜃₁,𝜃₂<𝑞⁻≤𝑞⁺, we have 1/𝑝⁺−1/𝑞⁺>0 and 𝛾_𝑘→∞. Thus, for sufficiently large 𝑘, we have 𝜑(𝑢)→∞ with 𝑢∈𝑍_𝑘 and ‖𝑢‖=𝛾_𝑘 as 𝑘→∞. In other cases, similarly, we can deduce 𝜑(𝑢)⟶∞,since𝛼_𝑘⟶0,𝑞_𝑘=0,𝑘⟶∞.(3.27) So (A2) holds.
By virtue of (𝑓₁) and (𝑔₁), there exist two positive constants 𝐶₄,𝐶₅ such that 𝐹(𝑥,𝑢)≥𝐶₄|𝑢|^𝜃1−1,∀(𝑥,𝑢)∈Ω×ℝ;𝐺(𝑥,𝑢)≥𝐶₅|𝑢|^𝜃2−1,∀(𝑥,𝑢)∈𝜕Ω×ℝ.(3.28) Letting 𝑢∈𝑌_𝑘, we have 𝜑(𝑢)≤1𝑝⁻Γ(𝑢)−𝑐¹𝑞⁺_𝜕Ω≤1|𝑢|^𝑞(𝑥)𝑑𝜎−𝜆_Ω𝐹(𝑥,𝑢)𝑑𝑥−𝜇_𝜕Ω𝐺(𝑥,𝑢)𝑑𝜎𝑝⁻max𝜉₂‖𝑢‖^𝑝+,𝜁₂‖𝑢‖^𝑝−−𝑐¹𝑞⁺−𝐶₅min|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}−𝐶⁴𝜆_Ω|𝑢|^𝜃1𝑑𝑥𝜇_𝜕Ω|𝑢|^𝜃2𝑑𝜎+𝐶₆.(3.29) If max{𝜉₂‖𝑢‖^𝑝+,𝜁₂‖𝑢‖^𝑝−}=𝜉₂‖𝑢‖^𝑝+,min{|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}}=|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}, then we have 𝜑(𝑢)≤𝜉²𝑝⁻‖𝑢‖^𝑝+−𝑐¹𝑞⁺|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}−𝐶₄𝜆_Ω|𝑢|^𝜃1𝑑𝑥−𝐶₅𝜇_𝜕Ω|𝑢|^𝜃2𝑑𝜎+𝐶₆.(3.30) Since dim𝑌_𝑘<∞, all norms are equivalent in 𝑌_𝑘. So we get 𝜑(𝑢)≤𝜉²𝑝⁻‖𝑢‖^𝑝+−𝑐¹𝑞⁺𝐶₇‖𝑢‖^𝑞−−𝐶₈𝜆‖𝑢‖^𝜃1−𝐶₉𝜇‖𝑢‖^𝜃2+𝐶₆.(3.31) Also, note that 𝑞⁻>𝜃₁,𝜃₂>𝑝⁺, Then, we get 𝜑(𝑢)→−∞ as ‖𝑢‖→∞. For other cases, the proofs are similar and we omit them here. So (A3) holds. From the proof of (A2) and (A3), we can choose 𝜌_𝑘>𝛾_𝑘>0. Thus, we complete the proof.

This time our idea is to show that 𝜑 possesses a nontrivial global minimum point in 𝑋.

Theorem 3.8. Let 𝛼⁺,𝛽⁺,𝑞⁺<𝑝⁻, and assume (𝑓₀), (𝑔₀) are satisfied; then (1.1) has a weak solution.

Proof. Firstly, we show that 𝜑 is coercive. For sufficiently large norm of 𝑢(‖𝑢‖≥1), and by virtue of (3.23), 𝜑(𝑢)=_Ω𝑎(𝑥)||∇𝑢||^𝑝(𝑥)+𝑏(𝑥)|𝑢|^𝑝(𝑥)𝑝(𝑥)𝑑𝑥−𝜆_Ω𝐹(𝑥,𝑢)𝑑𝑥−_𝜕Ω𝑐(𝑥)𝑞(𝑥)|𝑢|^𝑞(𝑥)−𝜇_𝜕Ω𝑑𝜎≥𝜉𝐺(𝑥,𝑢)𝑑𝜎₁𝑝⁺‖𝑢‖^𝑝−−||𝜆||_Ω𝐶₁1+|𝑢|^𝛼(𝑥)𝑑𝑥−𝑐²𝑞⁻_𝜕Ω≥𝜉|𝑢|^𝑞(𝑥)𝑑𝜎−||𝜇||_𝜕Ω𝐶₂1+|𝑢|^𝛽(𝑥)𝑑𝜎₁𝑝⁺‖𝑢‖^𝑝−−||𝜆||𝐶₁max|𝑢|^𝛼+_{𝐿𝛼(𝑥)(Ω)},|𝑢|^𝛼−_{𝐿𝛼(𝑥)(Ω)}−𝑐²𝑞⁻−||𝜇||𝐶₂max|𝑢|^𝑞+_{𝐿𝑞(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}max|𝑢|^𝛽+_{𝐿𝛽(𝑥)(𝜕Ω)},|𝑢|^𝛽−_{𝐿𝛽(𝑥)(𝜕Ω)}−𝐶¹⁰.(3.32) If mmaaxx||𝑢𝑢||^𝛼+_{𝐿𝛼(𝑥)(Ω)},_{𝑞+𝐿𝑞(𝑥)(𝜕Ω)}|𝑢|^𝛼−_{𝐿𝛼(𝑥)(Ω)}=|𝑢|^𝛼+_𝐿^𝛼(𝑥)_(Ω),max|𝑢|_{𝛽+𝐿𝛽(𝑥)(𝜕Ω)},|𝑢|^𝑞−_{𝐿𝑞(𝑥)(𝜕Ω)}=|𝑢|^𝑞+_𝐿^𝑞(𝑥)_(𝜕Ω),,|𝑢|^𝛽−_{𝐿𝛽(𝑥)(𝜕Ω)}=|𝑢|^𝛽+_𝐿^𝛽(𝑥)_(𝜕Ω),(3.33) then 𝜑(𝑢)≥𝜉¹𝑝⁺‖𝑢‖^𝑝−−𝐶₁₁||𝜆||‖𝑢‖^𝛼+−𝐶₁₂‖𝑢‖^𝑞+−𝐶₁₃||𝜇||‖𝑢‖^𝛽+−𝐶₁₀⟶∞as‖𝑢‖⟶∞.(3.34) So 𝜑 is coercive since 𝛼⁺,𝛽⁺,𝑞⁺<𝑝⁻. Secondly, by Proposition 2.2, it is easy to verify that 𝜑 is weakly lower semicontinuous. Thus, 𝜑 is bounded below and 𝜑 attains its infimum in 𝑋, that is, 𝜑(𝑢₀)=inf_𝑢∈𝑋𝜑(𝑢) and 𝑢₀ is a critical point of 𝜑, which is a weak solution of (1.1).

In the Theorem 3.8, we cannot guarantee that 𝑢₀ is nontrivial. In fact, under the assumptions on the above theorem, we can also get a nontrivial weak solution of 𝜑.

Corollary 3.9. Under the assumptions in Theorem 3.8, if one of the following conditions holds, (1.1) has a nontrivial weak solution.(1)If 𝜆,𝜇≠0, there exist two positive constants 𝑑₁,𝑑₂<𝑝⁻ such that liminf_𝑢→0sgn(𝜆)𝐹(𝑥,𝑢)|𝑢|^𝑑1liminf_𝑢→0>0,for𝑥∈Ωuniformly,sgn(𝜇)𝐺(𝑥,𝑢)|𝑢|^𝑑2>0,for𝑥∈𝜕Ωuniformly.(3.35)(2)If 𝜆=0,𝜇≠0, there exist two positive constants 𝑑₂<𝑝⁻ such that liminf_𝑢→0sgn(𝜇)𝐺(𝑥,𝑢)|𝑢|^𝑑2>0,for𝑥∈𝜕Ωuniformly.(3.36)(3)If 𝜆≠0,𝜇=0, there exist two positive constants 𝑑₁<𝑝⁻ such that lim_𝑢→0infsgn(𝜆)𝐹(𝑥,𝑢)|𝑢|^𝑑1>0,for𝑥∈Ωuniformly.(3.37)

Proof. From Theorem 3.8, we know that 𝜑 has a global minimum point 𝑢₀. We just need to show that 𝑢₀ is nontrivial. We only consider the case 𝜆,𝜇≠0 here. From (1), we know that for 0<𝑢<1 small enough, there exists two positive constants 𝐶₁₄,𝐶₁₅>0 such that sgn(𝜆)𝐹(𝑥,𝑢)≥𝐶₁₄|𝑢|^𝑑1,sgn(𝜇)𝐺(𝑥,𝑢)≥𝐶₁₅|𝑢|^𝑑2.(3.38) Choose 𝑢≡𝑀>0; then 𝑢∈𝑋. For 0<𝑡<1 small enough, we have 𝜑𝑡𝑢≤𝑏²𝑡^𝑝−𝑝⁻_Ω||𝑢||^𝑝(𝑥)𝑑𝑥−||𝜆||_Ωsgn(𝜆)𝐹𝑥,𝑡𝑢𝑑𝑥−𝑐¹𝑞⁺_𝜕Ω−||𝜇||||𝑡𝑢||^𝑞(𝑥)𝑑𝜎_𝜕Ω≤𝑏sgn(𝜇)𝐺𝑥,𝑡𝑢𝑑𝜎₂𝑡^𝑝−𝑝⁻_Ω||𝑀||^𝑝(𝑥)d𝑥−𝐶₁₄||𝜆||𝑡^𝑑1_Ω||𝑀||^𝑑1𝑑𝑥−𝑐¹𝑞⁺𝑡^𝑞−_𝜕Ω−𝐶₁₅||𝑀||^𝑞(𝑥)𝑑𝜎||𝜇||𝑡^𝑑2_𝜕Ω≤𝐶₁₆||𝑀||^𝑑2𝑑𝜎𝑡^𝑝−−𝐶₁₇||𝜆||𝑡^𝑑1−𝐶₁₈𝑐₁𝑡^𝑞−−𝐶₁₉||𝜇||𝑡^𝑑2.(3.39) Since 𝑑₁,𝑑₂,<𝑝⁻ and 𝑞⁻≤𝑞⁺<𝑝⁻, there exists 0<𝑡₀<1 small enough such that 𝜑(𝑡₀𝑢)<0. So the global minimum point 𝑢₀ of 𝜑 is nontrivial.

Remark 3.10. Suppose that 𝑓(𝑥,𝑢)=sgn(𝜆)|𝑢|^{𝛼(𝑥)−2}𝑢,𝑔(𝑥,𝑢)=sgn(𝜇)|𝑢|^{𝛽(𝑥)−2}𝑢 and 𝑝⁻>𝛼⁺,𝛽⁺,𝑞⁺; then the conditions in Corollary 3.9 can be fulfilled.