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, 𝐿𝑝(𝑥) and 𝑊1,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 𝐿𝑝(𝑥) are special cases of Orlicz spaces 𝐿𝜑 originated by Nakano [5] and developed by Musielak and Orlicz [6, 7], where 𝑓∈𝐿𝜑 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: ||||−divğ‘Ž(𝑥)∇𝑢𝑝(𝑥)−2∇𝑢+𝑏(𝑥)|𝑢|𝑝(𝑥)−2||||𝑢=𝜆𝑓(𝑥,𝑢),𝑥∈Ω,ğ‘Ž(𝑥)∇𝑢𝑝(𝑥)−2𝜕𝑢𝜕𝜈=𝑐(𝑥)|𝑢|ğ‘ž(𝑥)−2𝑢+𝜇𝑔(𝑥,𝑢),𝑥∈𝜕Ω,(1.1) where Ω⊂ℝ𝑛 is a bounded domain with Lipschitz boundary 𝜕Ω,𝜕/𝜕𝜈 is outer unit normal derivative, 𝑝(𝑥)∈𝐶(Ω),ğ‘ž(𝑥)∈𝐶(𝜕Ω),𝑝(𝑥),ğ‘ž(𝑥)>1, and 𝑝(𝑥)â‰ ğ‘ž(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 ğ‘Ž(𝑥),𝑏(𝑥), and 𝑐(𝑥) satisfy 0<ğ‘Ž1â‰¤ğ‘Ž(𝑥)â‰¤ğ‘Ž2,0<𝑏1≤𝑏(𝑥)≤𝑏2, and 0≤𝑐1≤𝑐(𝑥)≤𝑐2.

The operator −Δ𝑝(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,ğ‘Ž(𝑥)=𝑏(𝑥)=𝑐(𝑥)≡1,𝑝(𝑥)≡𝑝, and ğ‘ž(𝑥)â‰¡ğ‘ž (a constant), then problem (1.1) becomes ||||−div∇𝑢𝑝−2∇𝑢+|𝑢|𝑝−2||||𝑢=𝑓(𝑥,𝑢),𝑥∈Ω,∇𝑢𝑝−2𝜕𝑢𝜕𝜈=|𝑢|ğ‘žâˆ’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=ğ‘ž 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 ğ‘ž(𝑥)â‰¡ğ‘ž (a constant), then problem (1.1) becomes ||||−divğ‘Ž(𝑥)∇𝑢𝑝−2∇𝑢+𝑏(𝑥)|𝑢|𝑝−2||||𝑢=𝑓(𝑥,𝑢),𝑥∈Ω,ğ‘Ž(𝑥)∇𝑢𝑝−2𝜕𝑢𝜕𝜈=𝑐(𝑥)|𝑢|ğ‘žâˆ’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 𝑊1,𝑝(𝑥)(Ω), which we call variable exponent Sobolev spaces. We state some basic properties of the spaces 𝑊1,𝑝(𝑥)(Ω), which will be used later (for more details, see [25, 26]). Let Ω be a bounded domain of ℝ𝑛, and denote 𝐶+Ω=𝑝(𝑥)∣𝑝(𝑥)∈𝐶Ω;𝑝(𝑥)>1,∀𝑥∈Ω.(2.1) For 𝑝(𝑥)∈𝐶+(Ω) write 𝑝+=max𝑥∈Ω𝑝(𝑥),𝑝−=min𝑥∈Ω𝑝(𝑥).(2.2) We can also denote 𝐶+(𝜕Ω) and ğ‘ž+,ğ‘žâˆ’ for any ğ‘ž(𝑥)∈𝐶(𝜕Ω), and define 𝐿𝑝(𝑥)(Ω)=𝑢∣𝑢isameasurablereal-valuedfunction,Ω||||𝑢(𝑥)𝑝(𝑥),𝐿𝑑𝑥<âˆžğ‘(𝑥)(𝜕Ω)=𝑢∣𝑢∶𝜕Ω⟶ℝisameasurablereal-valuedfunction,𝜕Ω||||𝑢(𝑥)𝑝(𝑥),ğ‘‘ğœŽ<∞(2.3) with norms on 𝐿𝑝(𝑥)(Ω) and 𝐿𝑝(𝑥)(𝜕Ω) defined by|𝑢|𝐿𝑝(𝑥)(Ω)=|𝑢|𝑝(𝑥)=inf𝜆>0∶Ω|||𝑢(𝑥)𝜆|||𝑝(𝑥),𝑑𝑥≤1|𝑢|𝐿𝑝(𝑥)(𝜕Ω)=inf𝜏>0∶𝜕Ω|||𝑢(𝑥)𝜏|||𝑝(𝑥),ğ‘‘ğœŽâ‰¤1(2.4) where ğ‘‘ğœŽ is the surface measure on 𝜕Ω. Then, (𝐿𝑝(𝑥)(Ω),|⋅|𝑝(𝑥)) and (𝐿𝑝(𝑥)(𝜕Ω),|⋅|𝐿𝑝(𝑥)(𝜕Ω)) become Banach spaces, which we call variable exponent Lebesgue spaces. Let us define the space 𝑊1,𝑝(𝑥)(Ω)=𝑢∈𝐿𝑝(𝑥)||||(Ω)∶∇𝑢∈𝐿𝑝(𝑥),(Ω)(2.5) equipped with the norm ‖𝑢‖=inf𝜆>0∶Ω|||∇𝑢(𝑥)𝜆|||𝑝(𝑥)+|||𝑢(𝑥)𝜆|||𝑝(𝑥).𝑑𝑥≤1(2.6) For 𝑢∈𝑊1,𝑝(𝑥)(Ω), if we define â€–ğ‘¢â€–î…žî‚»î€œ=inf𝜆>0∶Ω|||ğ‘Ž(𝑥)∇𝑢(𝑥)𝜆|||𝑝(𝑥)|||+𝑏(𝑥)𝑢(𝑥)𝜆|||𝑝(𝑥),𝑑𝑥≤1(2.7) then, from the assumptions of ğ‘Ž(𝑥) and 𝑏(𝑥), it is easy to check that â€–ğ‘¢â€–î…ž is an equivalent norm on 𝑊1,𝑝(𝑥)(Ω). For simplicity, we denote Γ(𝑢)=Ω||||ğ‘Ž(𝑥)∇𝑢𝑝(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 𝑊01,𝑝(𝑥)(Ω) the closure of ğ¶âˆž0(Ω) in 𝑊1,𝑝(𝑥)(Ω).

Proposition 2.1 (see [21, 28]). (1) The space (𝐿𝑝(𝑥)(Ω), |⋅|𝑝(𝑥)) is a separable, uniformly convex Banach space, and its conjugate space is ğ¿ğ‘ž(𝑥)(Ω), where 1/ğ‘ž(𝑥)+1/𝑝(𝑥)=1. For any 𝑢∈𝐿𝑝(𝑥)(Ω) and ğ‘£âˆˆğ¿ğ‘ž(𝑥)(Ω), one has ||||Ω||||≤1𝑢𝑣d𝑥𝑝−+1ğ‘žâˆ’î‚¶|𝑢|𝑝(𝑥)|𝑣|ğ‘ž(𝑥).(2.9)

(2) If 𝑝1,𝑝2∈𝐶+(Ω), 𝑝1(𝑥)≤𝑝2(𝑥), for any 𝑥∈Ω, then 𝐿𝑝2(𝑥)(Ω)↪𝐿𝑝1(𝑥)(Ω) and the imbedding is continuous.

Proposition 2.2 (see [20, 21, 28]). (1)   𝑊1,𝑝(𝑥)(Ω),𝑊01,𝑝(𝑥)(Ω) are separable reflexive Banach spaces.
(2) If ğ‘ž(𝑥)∈𝐶+(Ω) and ğ‘ž(𝑥)<𝑝∗(𝑥) for any 𝑥∈Ω, then the embedding from 𝑊1,𝑝(𝑥)(Ω) into ğ¿ğ‘ž(𝑥)(Ω) is compact and continuous, where 𝑝∗(âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¥)=𝑛𝑝(𝑥)𝑛−𝑝(𝑥),if𝑝(𝑥)<𝑛,∞,if𝑝(𝑥)≥𝑛.(2.10)
(3) If ğ‘ž(𝑥)∈𝐶+(𝜕Ω) and ğ‘ž(𝑥)<𝑝∗(𝑥) for any 𝑥∈𝜕Ω, then the trace imbedding from 𝑊1,𝑝(𝑥)(Ω) into ğ¿ğ‘ž(𝑥)(𝜕Ω) is compact and continuous, where 𝑝∗(âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¥)=(𝑛−1)𝑝(𝑥)𝑛−𝑝(𝑥),if𝑝(𝑥)<𝑛,∞,if𝑝(𝑥)≥𝑛.(2.11)
(4) (Poincaré inequality) There is a constant 𝐶>0, such that |𝑢|𝑝(𝑥)||||≤𝐶∇𝑢𝑝(𝑥)∀𝑢∈𝑊01,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)â‰¤ğ‘Ž(𝑥)+𝑏|𝑠|𝑝1(𝑥)/𝑝2(𝑥),forany𝑥∈Ω,𝑠∈ℝ,(2.13) where 𝑝1(𝑥), 𝑝2(𝑥)∈𝐶+(Ω), ğ‘Ž(𝑥)∈𝐿𝑝2(𝑥)(Ω), ğ‘Ž(𝑥)≥0, and 𝑏≥0 is a constant, then the Nemytsky operator from 𝐿𝑝1(𝑥)(Ω) to 𝐿𝑝2(𝑥)(Ω) defined by (𝑁𝑓(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∈𝐿𝑝(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)dğœŽ,∀𝑢∈𝐿𝑝(𝑥)(𝜕Ω).(2.15) Then, (1)|𝑢|𝐿𝑝(𝑥)(𝜕Ω)>1 implies |𝑢|𝑝−𝐿𝑝(𝑥)(𝜕Ω)≤𝜌(𝑢)≤|𝑢|𝑝+𝐿𝑝(𝑥)(𝜕Ω),(2)|𝑢|𝐿𝑝(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 𝑊1,𝑝(𝑥)(Ω),  𝑋∗ denote the dual space of 𝑊1,𝑝(𝑥)(Ω), ⟨⋅⟩ denote the dual pair, and let → represent strong convergence, ⇀ represent weak convergence, 𝐶, 𝐶𝑖 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 ğ‘î…ž1≥0,ğ‘î…ž2>0 such that ||||𝑔(𝑥,𝑢)â‰¤ğ‘î…ž1+ğ‘î…ž2|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)=Ω||||ğ‘Ž(𝑥)∇𝑢𝑝(𝑥)+𝑏(𝑥)|𝑢|𝑝(𝑥)𝑝(𝑥)d𝑥−𝜆Ω−𝐹(𝑥,𝑢)𝑑𝑥𝜕Ω𝑐(𝑥)|ğ‘ž(𝑥)𝑢|ğ‘ž(𝑥)dğœŽâˆ’ğœ‡ğœ•Î©ğº(𝑥,𝑢)ğ‘‘ğœŽ,(3.5) where 𝐹(𝑥,𝑢) and 𝐺(𝑥,𝑢) are denoted by 𝐹(𝑥,𝑢)=𝑢0𝑓(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, î«ğœ‘î…ž(=𝑢),𝑣Ω||||ğ‘Ž(𝑥)∇𝑢𝑝(𝑥)−2∇𝑢∇𝑣+𝑏(𝑥)|𝑢|𝑝(𝑥)−2𝑢𝑣𝑑𝑥−𝜆Ω−𝑓(𝑥,𝑢)𝑣𝑑𝑥𝜕Ω𝑐(𝑥)|𝑢|ğ‘ž(𝑥)−2î€œğ‘¢ğ‘£ğ‘‘ğœŽâˆ’ğœ‡ğœ•Î©ğ‘”(𝑥,𝑢)ğ‘£ğ‘‘ğœŽ,(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)=Ω||||ğ‘Ž(𝑥)∇𝑢𝑝(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 î«ğ¼î…ž(=𝑢),𝑣Ω||||ğ‘Ž(𝑥)∇𝑢𝑝(𝑥)−2∇𝑢∇𝑣+𝑏(𝑥)|𝑢|𝑝(𝑥)−2𝑢𝑣d𝑥,∀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 𝑢𝑛⇀𝑢 in 𝑋 and limsupğ‘›â†’âˆžâŸ¨ğ¼â€²(𝑢𝑛)−𝐼′(𝑢),𝑢𝑛−𝑢⟩≤0, then 𝑢𝑛→𝑢 in 𝑋,(iii)𝐼′∶𝑋→𝑋∗ is a homeomorphism.

Proposition 3.2 (see [19]). If one denotes 𝐽(𝑢)=𝜕Ω𝑐(𝑥)|ğ‘ž(𝑥)𝑢|ğ‘ž(𝑥)dğœŽ,∀𝑢∈𝑋,(3.10) where ğ‘ž(𝑥)∈𝐶+(𝜕Ω) and ğ‘ž(𝑥)<𝑝∗(𝑥) for any 𝑥∈𝜕Ω, then 𝐽∈𝐶1(𝑋,ℝ) and the derivative operator 𝐽′ of 𝐽 is î«ğ½î…ž(=𝑢),𝑣𝜕Ω𝑐(𝑥)|𝑢|ğ‘ž(𝑥)−2𝑢𝑣dğœŽ,∀𝑢,𝑣∈𝑋,(3.11) and one has that 𝐽∶𝑋→ℝ and 𝐽′∶𝑋→𝑋∗ are sequentially weakly-strongly continuous, namely, 𝑢𝑛⇀𝑢 in 𝑋 implies 𝐽′(𝑢𝑛)→𝐽′(𝑢).

Let 𝑋 be a reflexive and separable Banach space. There exist 𝑒𝑖∈𝑋 and 𝑒∗𝑗∈𝑋∗ such that 𝑋=𝑒span𝑖∶𝑖=1,2,…,𝑋∗=𝑒span∗𝑗,𝑒∶𝑗=1,2,…𝑖,𝑒∗𝑗=1,𝑖=𝑗,0,𝑖≠𝑗.(3.12) For 𝑘=1,2,…, denote 𝑋𝑘𝑒=span𝑘,𝑌𝑘=𝑘𝑖=1𝑋𝑖,𝑍𝑘=𝑖≥𝑘𝑋𝑖.(3.13)

One important aspect of applying the standard methods of variational theory is to show that the functional 𝜑 satisfies the ğ‘ƒğ‘Žğ‘™ğ‘Žğ‘–ğ‘ -ğ‘†ğ‘šğ‘Žğ‘™ğ‘’ 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 {𝑢𝑛}⊂𝑋 such that 𝜑𝑢𝑛⟶𝑐,ğœ‘î…žî€·ğ‘¢ğ‘›î€¸âŸ¶0in𝑋∗,asğ‘›âŸ¶âˆž(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 ğ‘ƒğ‘Žğ‘™ğ‘Žğ‘–ğ‘ -ğ‘†ğ‘šğ‘Žğ‘™ğ‘’ 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 𝑋𝑘,𝑌𝑘 and 𝑍𝑘 are defined by (3.13).Suppose that, for every 𝑘∈𝐍, there exist 𝜌𝑘>𝛾𝑘>0 such that(A2)inf𝑢∈𝑍𝑘,‖𝑢‖=𝛾𝑘𝜑(𝑢)→∞ as ğ‘˜â†’âˆž,(A3)max𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘𝜑(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 ğ‘ž(𝑥)∈𝐶+(𝜕Ω),ğ‘ž(𝑥)<𝑝∗(𝑥),forall𝑥∈𝜕Ω, denote 𝛼𝑘=sup|𝑢|𝐿𝛼(𝑥)(Ω)∶‖𝑢‖=1,𝑢∈𝑍𝑘;ğ‘žğ‘˜î€½=sup|𝑢|ğ¿ğ‘ž(𝑥)(𝜕Ω)∶‖𝑢‖=1,𝑢∈𝑍𝑘,(3.15) then limğ‘˜â†’âˆžğ›¼ğ‘˜=0,limğ‘˜â†’âˆžğ‘žğ‘˜=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,ğ‘žâˆ’>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)=𝜆Ω𝐹(𝑥,𝑢)𝑑𝑥+𝜇𝜕Ω𝐺(𝑥,𝑢)ğ‘‘ğœŽ,(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 {𝑢𝑛}⊂𝑋 such that 𝜑𝑢𝑛⟶𝑐,ğœ‘î…žî€·ğ‘¢ğ‘›î€¸âŸ¶0,in𝑋∗,asğ‘›âŸ¶âˆž.(3.17) Then, for 𝑛 large enough, we can find 𝑀3>0 such that ||𝜑𝑢𝑛||≤𝑀3.(3.18) Since 𝜑′(𝑢𝑛)→0, we have ⟨𝜑′(𝑢𝑛),𝑢𝑛⟩→0. In particular, {⟨𝜑′(𝑢𝑛),𝑢𝑛⟩} is bounded. Thus, there exists 𝑀4>0 such that ||î«ğœ‘î…žî€·ğ‘¢ğ‘›î€¸,𝑢𝑛||≤𝑀4.(3.19) We claim that the sequence {𝑢𝑛} is bounded. If it is not true, by passing a subsequence if necessary, we may assume that ‖𝑢𝑛‖→+∞. Without loss of generality, we assume that ‖𝑢𝑛‖≥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<ğ‘žâˆ’, we have 𝑀3𝑢≥𝜑𝑛𝑢=𝐼𝑛𝑢−𝐽𝑛𝑢−Φ𝑛≥1𝑝+Γ𝑢𝑛−1ğ‘žâˆ’î€œğœ•Î©ğ‘||𝑢(𝑥)𝑛||ğ‘ž(𝑥)î€·ğ‘¢ğ‘‘ğœŽâˆ’Î¦ğ‘›î€¸,≥1𝑝+Γ𝑢𝑛−1𝜃𝜕Ω𝑐||𝑢(𝑥)𝑛||ğ‘ž(𝑥)î€·ğ‘¢ğ‘‘ğœŽâˆ’Î¦ğ‘›î€¸,𝑀(3.20)4î«ğœ‘â‰¥âˆ’î…žî€·ğ‘¢ğ‘›î€¸,𝑢𝑛𝑢=−Γ𝑛+𝜕Ω𝑐||𝑢(𝑥)𝑛||ğ‘ž(𝑥)î«Î¦ğ‘‘ğœŽ+î…žî€·ğ‘¢ğ‘›î€¸,𝑢𝑛.(3.21) By virtue of assumptions (𝑓1) and (𝑔1) and combining (3.20) and (3.21), we have 𝜃𝑀3+𝑀4≥𝜃𝑝+Γ𝑢−1𝑛𝑢−𝜃Φ𝑛+î«Î¦î…žî€·ğ‘¢ğ‘›î€¸,𝑢𝑛≥𝜃𝑝+𝜉−11‖‖𝑢𝑛‖‖𝑝−+𝜆Ω𝑓𝑥,𝑢𝑛𝑢𝑛−𝜃𝐹𝑥,𝑢𝑛𝑑𝑥+𝜇𝜕Ω𝑔𝑥,𝑢𝑛𝑢𝑛−𝜃𝐺𝑥,ğ‘¢ğ‘›â‰¥î‚µğœƒî€¸î€¸ğ‘‘ğœŽğ‘+𝜉−11‖‖𝑢𝑛‖‖𝑝−−𝐶.(3.22) Note that 𝜃=min{𝜃1,𝜃2}>𝑝+, let ğ‘›â†’âˆž we obtian a contradiction. It follows that the sequence {𝑢𝑛} 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+,ğ‘žâˆ’>𝜃1,𝜃2, and 𝜆,𝜇≥0, and assumed that (𝑓0)−(𝑓2),(𝑔0)−(𝑔2) are satisfied; then 𝜑 has a sequence of critical points {±𝑢𝑛} such that 𝜑(±𝑢𝑛)→∞ as ğ‘›â†’âˆž.

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 𝜌𝑘>𝛾𝑘>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∈𝑍𝑘 with ‖𝑢‖>1 appropriately large such that 𝜉1‖𝑢‖𝑝−<𝜁1‖𝑢‖𝑝+, we have ≥1𝜑(𝑢)=𝐼(𝑢)−𝐽(𝑢)−Φ(𝑢)𝑝+𝑐Γ(𝑢)−2ğ‘žâˆ’î€œğœ•Î©|𝑢|ğ‘ž(𝑥)î€œğ‘‘ğœŽâˆ’ğœ†Î©ğ¶11+|𝑢|𝛼(𝑥)𝑑𝑥−𝜇𝜕Ω𝐶21+|𝑢|𝛽(𝑥)≥1ğ‘‘ğœŽğ‘+𝜉min1‖𝑢‖𝑝−,𝜁1‖𝑢‖𝑝+−𝑐2ğ‘žâˆ’î‚†max|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω)−𝜆𝐶1max|𝑢|𝛼+𝐿𝛼(𝑥)(Ω),|𝑢|𝛼−𝐿𝛼(𝑥)(Ω)−𝜇𝐶2max|𝑢|𝛽+𝐿𝛽(𝑥)(𝜕Ω),|𝑢|𝛽−𝐿𝛽(𝑥)(𝜕Ω)−𝐶3≥𝜉1𝑝+‖𝑢‖𝑝−−𝐶(ğ‘žâˆ’î‚†,𝜆,𝜇)max|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|𝛼+𝐿𝛼(𝑥)(Ω),|𝑢|𝛼−𝐿𝛼(𝑥)(Ω),|𝑢|𝛽+𝐿𝛽(𝑥)(𝜕Ω),|𝑢|𝛽−𝐿𝛽(𝑥)(𝜕Ω)−𝐶3.(3.24) If max{|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|𝛼+𝐿𝛼(𝑥)(Ω),|𝑢|𝛼−𝐿𝛼(𝑥)(Ω),|𝑢|𝛽+𝐿𝛽(𝑥)(𝜕Ω),|𝑢|𝛽−𝐿𝛽(𝑥)(𝜕Ω)}=|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω), then by Proposition 3.5, we have 𝜑𝑢𝑛≥𝜉1𝑝+‖𝑢‖𝑝−−𝐶(ğ‘žâˆ’,𝜆,𝜇)|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω)−𝐶3≥𝜉1𝑝+‖𝑢‖𝑝−−𝐶(ğ‘žâˆ’,𝜆,𝜇)ğ‘žğ‘ž+ğ‘˜â€–ğ‘¢â€–ğ‘ž+−𝐶3.(3.25) Choose 𝛾𝑘=(ğ‘ž+𝐶(ğ‘žâˆ’,𝜆,𝜇)(ğ‘ž_𝑘(ğ‘ž+))/𝜉_1)1/(ğ‘âˆ’âˆ’ğ‘ž+). For 𝑢∈𝑍𝑘 with ‖𝑢‖=𝛾𝑘, we have 𝜑(𝑢)≥𝜉11𝑝+−1ğ‘ž+𝛾𝑝−𝑘−𝐶3.(3.26) Since ğ‘žğ‘˜â†’0 as ğ‘˜â†’âˆž and 1<𝑝−≤𝑝+<𝜃1,𝜃2<ğ‘žâˆ’â‰¤ğ‘ž+, we have 1/𝑝+−1/ğ‘ž+>0 and ğ›¾ğ‘˜â†’âˆž. Thus, for sufficiently large 𝑘, we have 𝜑(𝑢)→∞ with 𝑢∈𝑍𝑘 and ‖𝑢‖=𝛾𝑘 as ğ‘˜â†’âˆž. In other cases, similarly, we can deduce 𝜑(𝑢)⟶∞,since𝛼𝑘⟶0,ğ‘žğ‘˜=0,ğ‘˜âŸ¶âˆž.(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∈𝑌𝑘, we have 1𝜑(𝑢)≤𝑝−𝑐Γ(𝑢)−1ğ‘ž+𝜕Ω|𝑢|ğ‘ž(𝑥)î€œğ‘‘ğœŽâˆ’ğœ†Î©î€œğ¹(𝑥,𝑢)𝑑𝑥−𝜇𝜕Ω≤1𝐺(𝑥,𝑢)ğ‘‘ğœŽğ‘âˆ’î‚†ğœ‰max2‖𝑢‖𝑝+,𝜁2‖𝑢‖𝑝−−𝑐1ğ‘ž+|min𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω)−𝐶4𝜆Ω|𝑢|𝜃1𝑑𝑥−𝐶5𝜇𝜕Ω|𝑢|𝜃2ğ‘‘ğœŽ+𝐶6.(3.29) If max{𝜉2‖𝑢‖𝑝+,𝜁2‖𝑢‖𝑝−}=𝜉2‖𝑢‖𝑝+,min{|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω)}=|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω), then we have 𝜉𝜑(𝑢)≤2𝑝−‖𝑢‖𝑝+−𝑐1ğ‘ž+|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω)−𝐶4𝜆Ω|𝑢|𝜃1𝑑𝑥−𝐶5𝜇𝜕Ω|𝑢|𝜃2ğ‘‘ğœŽ+𝐶6.(3.30) Since dim𝑌𝑘<∞, all norms are equivalent in 𝑌𝑘. So we get 𝜉𝜑(𝑢)≤2𝑝−‖𝑢‖𝑝+−𝑐1ğ‘ž+𝐶7â€–ğ‘¢â€–ğ‘žâˆ’âˆ’ğ¶8𝜆‖𝑢‖𝜃1−𝐶9𝜇‖𝑢‖𝜃2+𝐶6.(3.31) Also, note that ğ‘žâˆ’>𝜃1,𝜃2>𝑝+, Then, we get 𝜑(𝑢)→−∞ as â€–ğ‘¢â€–â†’âˆž. 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 𝜌𝑘>𝛾𝑘>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+,ğ‘ž+<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)=Ω||||ğ‘Ž(𝑥)∇𝑢𝑝(𝑥)+𝑏(𝑥)|𝑢|𝑝(𝑥)𝑝(𝑥)𝑑𝑥−𝜆Ω𝐹(𝑥,𝑢)𝑑𝑥−𝜕Ω𝑐(𝑥)ğ‘ž(𝑥)|𝑢|ğ‘ž(𝑥)î€œğ‘‘ğœŽâˆ’ğœ‡ğœ•Î©â‰¥ğœ‰ğº(𝑥,𝑢)ğ‘‘ğœŽ1𝑝+‖𝑢‖𝑝−−||𝜆||Ω𝐶11+|𝑢|𝛼(𝑥)𝑐𝑑𝑥−2ğ‘žâˆ’î€œğœ•Î©|𝑢|ğ‘ž(𝑥)||𝜇||î€œğ‘‘ğœŽâˆ’ğœ•Î©ğ¶21+|𝑢|𝛽(𝑥)î€¸â‰¥ğœ‰ğ‘‘ğœŽ1𝑝+‖𝑢‖𝑝−−||𝜆||𝐶1max|𝑢|𝛼+𝐿𝛼(𝑥)(Ω),|𝑢|𝛼−𝐿𝛼(𝑥)(Ω)−𝑐2ğ‘žâˆ’î‚†max|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω)−||𝜇||𝐶2max|𝑢|𝛽+𝐿𝛽(𝑥)(𝜕Ω),|𝑢|𝛽−𝐿𝛽(𝑥)(𝜕Ω)−𝐶10.(3.32) If max|𝑢|𝛼+𝐿𝛼(𝑥)(Ω),|𝑢|𝛼−𝐿𝛼(𝑥)(Ω)=|𝑢|𝛼+𝐿𝛼(𝑥)(Ω),max|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),|𝑢|ğ‘žâˆ’ğ¿ğ‘ž(𝑥)(𝜕Ω)=|𝑢|ğ‘ž+ğ¿ğ‘ž(𝑥)(𝜕Ω),max|𝑢|𝛽+𝐿𝛽(𝑥)(𝜕Ω),|𝑢|𝛽−𝐿𝛽(𝑥)(𝜕Ω)=|𝑢|𝛽+𝐿𝛽(𝑥)(𝜕Ω),(3.33) then 𝜉𝜑(𝑢)≥1𝑝+‖𝑢‖𝑝−−𝐶11||𝜆||‖𝑢‖𝛼+−𝐶12â€–ğ‘¢â€–ğ‘ž+−𝐶13||𝜇||‖𝑢‖𝛽+−𝐶10⟶∞asâ€–ğ‘¢â€–âŸ¶âˆž.(3.34) So 𝜑 is coercive since 𝛼+,𝛽+,ğ‘ž+<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)=inf𝑢∈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 liminf𝑢→0sgn(𝜆)𝐹(𝑥,𝑢)|𝑢|𝑑1>0,for𝑥∈Ωuniformly,liminf𝑢→0sgn(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 liminf𝑢→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 lim𝑢→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≤𝑏2𝑡𝑝−𝑝−Ω||𝑢||𝑝(𝑥)||𝜆||𝑑𝑥−Ωsgn(𝜆)𝐹𝑥,𝑡𝑢𝑐𝑑𝑥−1ğ‘ž+𝜕Ω||𝑡𝑢||ğ‘ž(𝑥)−||𝜇||î€œğ‘‘ğœŽğœ•Î©î€·sgn(𝜇)𝐺𝑥,ğ‘¡ğ‘¢î€¸â‰¤ğ‘ğ‘‘ğœŽ2𝑡𝑝−𝑝−Ω||𝑀||𝑝(𝑥)d𝑥−𝐶14||𝜆||𝑡𝑑1Ω||𝑀||𝑑1𝑐𝑑𝑥−1ğ‘ž+ğ‘¡ğ‘žâˆ’î€œğœ•Î©||𝑀||ğ‘ž(𝑥)ğ‘‘ğœŽâˆ’ğ¶15||𝜇||𝑡𝑑2𝜕Ω||𝑀||𝑑2ğ‘‘ğœŽâ‰¤ğ¶16𝑡𝑝−−𝐶17||𝜆||𝑡𝑑1−𝐶18𝑐1ğ‘¡ğ‘žâˆ’âˆ’ğ¶19||𝜇||𝑡𝑑2.(3.39) Since 𝑑1,𝑑2,<𝑝− and ğ‘žâˆ’â‰¤ğ‘ž+<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)−2𝑢,𝑔(𝑥,𝑢)=sgn(𝜇)|𝑢|𝛽(𝑥)−2𝑢 and 𝑝−>𝛼+,𝛽+,ğ‘ž+; then the conditions in Corollary 3.9 can be fulfilled.