Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces

Abstract

In this paper, by means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations of 𝑝(𝑥)-Kirchhoff type with nonstandard growth condition.

1. Introduction

Let Ω be smooth bounded domain of ℝ^𝑛,𝑛≥2, and let Ω₁⊂Ω be a subdomain with smooth boundary Σ satisfying Ω₁⊂Ω. Writing Γ=𝜕Ω and Ω₂=Ω⧵Ω₁ we have Ω=Ω₁∪Ω₂ and 𝜕Ω₂=Σ∪Γ (Figure 1).

Figure 1 

We are concerned with the existence of positive solutions to the following system of nonlinear elliptic equations:−𝑀_Ω11𝑝(𝑥)|∇𝑢|^𝑝(𝑥)𝑑𝑥div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢=𝑓(𝑥,𝑢),inΩ₁,−𝑁_Ω21𝑝(𝑥)|∇𝜐|^𝑝(𝑥)𝑑𝑥div|∇𝜐|^{𝑝(𝑥)−2}∇𝜐=𝑔(𝑥,𝜐),inΩ₂,𝜐=0,onΓ,(P) with the transmission conditions𝑀_Ω1|∇𝑢|^𝑝(𝑥)𝑑𝑥𝜕𝑢𝜕𝜂=𝑁_Ω2|∇𝜐|^𝑝(𝑥)𝑑𝑥𝜕𝜐𝜕𝜂,onΣ𝑢=𝜐,onΣ,(1.1) where 𝑀  and 𝑁 are positive continuous functions, 𝜂 is outward normal to Ω₂ and is inward to Ω₁, and 𝑝(𝑥)∈𝐶(Ω). By means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations of 𝑝(𝑥)-Kirchhoff type with nonstandard growth condition. We investigate the problem (P) when 𝑓(𝑥,𝑢)=𝜆₁|𝑢|^{𝑞(𝑥)−2}𝑢, 𝑔(𝑥,𝜐)=𝜆₂|𝜐|^{𝑞(𝑥)−2}𝜐, where 𝜆₁,𝜆₂>0 and 𝑝(𝑥), 𝑞(𝑥)∈𝐶(Ω) such that 1<𝑞(𝑥)<𝑝^∗(𝑥) for all 𝑥∈Ω, where 𝑝^∗(𝑥)=𝑛𝑝(𝑥)/𝑛−𝑝(𝑥) if 𝑝(𝑥)<𝑛 or 𝑝^∗(𝑥)=+∞ if 𝑝(𝑥)≥𝑛.

The operator −div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) is called the 𝑝(𝑥)-Laplacian which is a natural generalization of the 𝑝-Laplacian when 𝑝(𝑥)≡𝑝>1 (a constant). The 𝑝(𝑥)-Laplacian possesses more complicated nonlinearities than the 𝑝-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1, 2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [3].

Transmission problems arise in several applications in physics and biology. For instance, one of the important problems of the electrodynamics of solid media is the electromagnetic process research in ferromagnetic media with different dielectric constants. These problems appear as well as in solid mechanics if a body consists of composite materials. The existence and regularity results for linear transmission problems are well known, and a complete study can be found in [4]. We refer the reader to [5] for nonlinear elliptic transmission problems, to [6] for a nonlinear nonlocal elliptic transmission problem. Furthermore, uniqueness and regularity of the solutions to the thermoelastic transmission problem were investigated in [7].

We note that problem (P) with the transmission condition is a generalization of the stationary problem of two wave equations of Kirchhoff type,𝑢_𝑡𝑡−𝑀_Ω1|∇𝑢|²𝑑𝑥Δ₂𝑢=𝑓(𝑥,𝑢)inΩ₁,𝜐_𝑡𝑡−𝑁_Ω2|∇𝜐|²𝑑𝑥Δ₂𝜐=𝑔(𝑥,𝜐)inΩ₂,(1.2) which models the transverse vibrations of the membrane composed by two different materials in Ω₁ and Ω₂. Controllability and stabilization of transmission problems for the wave equations can be found in [8, 9]. We refer the reader to [10] for the stationary problems of Kirchhoff type, to [11] for elliptic equation 𝑝-Kirchhoff type, and to [12, 13] for 𝑝(𝑥)-Kirchhoff type equation.

2. Auxiliary Results

We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces 𝐿^𝑝(𝑥)(Ω), 𝑊^1,𝑝(𝑥)(Ω), and 𝑊₀^1,𝑝(𝑥)(Ω). In that context we refer to [14, 15] for the fundamental properties of these spaces.

By Ω we always denote a nonempty open subset of ℝ^𝑛. Set𝐶₊Ω=ℎ;ℎ∈𝐶Ω,ℎ(𝑥)>1∀𝑥∈Ω.(2.1) For any ℎ∈𝐶₊(Ω) we defineℎ₁∶=inf_𝑥∈Ωℎ(𝑥),ℎ₂∶=sup_𝑥∈Ωℎ(𝑥).(2.2) We define the variable exponent Lebesgue space 𝐿^𝑝(⋅)(Ω) to consist of all measurable functions 𝑢∶Ω→ℝ for which the modular𝜌_{𝑝(𝑥),Ω}(𝑢)=_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥(2.3) is finite. We define the Luxemburg norm on this space by the formula‖𝑢‖_{𝑝(𝑥),Ω}=inf𝛿>0∶𝜌_{𝑝(𝑥),Ω}𝑢𝛿≤1.(2.4) Equipped with this norm, 𝐿^𝑝(⋅)(Ω) is a separable and reflexive Banach space. Define the variable exponent Sobolev space 𝑊^1,𝑝(𝑥)(Ω) by𝑊^1,𝑝(𝑥)(Ω)=𝑢∈𝐿^𝑝(𝑥)(Ω);|∇𝑢|∈𝐿^𝑝(𝑥)(Ω),(2.5) and the norm‖𝑢‖_{1,𝑝(𝑥),Ω}=‖𝑢‖_{𝑝(𝑥),Ω}+‖∇𝑢‖_{𝑝(𝑥),Ω},∀𝑢∈𝑊^1,𝑝(𝑥)(Ω)(2.6) makes 𝑊^1,𝑝(𝑥)(Ω) a separable and reflexive Banach space. The space 𝑊₀^1,𝑝(𝑥)(Ω) is denoted by the closure of 𝐶^∞₀(Ω) in 𝑊^1,𝑝(𝑥)(Ω). 𝑊₀^1,𝑝(𝑥)(Ω) is a separable and reflexive Banach space.

Proposition 2.1 (see [14, 15]). Let 𝑝∈𝐶₊(Ω). Then conjugate space of 𝐿^𝑝(𝑥)(Ω) is 𝐿^𝑞(𝑥)(Ω), where 1/𝑝(𝑥)+1/𝑞(𝑥)=1. For any 𝑢∈𝐿^𝑝(𝑥)(Ω) and 𝜐∈ 𝐿^𝑞(𝑥)(Ω), one has 𝑢𝜐𝑑𝑥≤2‖_Ω𝑢‖_{𝑝(𝑥),Ω}‖𝜐‖_{𝑞(𝑥),Ω}.(2.7)

The next proposition illuminates the close relation between the ‖⋅‖_{𝑝(𝑥),Ω} and the convex modular 𝜌_{𝑝(𝑥),Ω}.

Proposition 2.2 (see [14, 15]). If 𝑢∈𝐿^𝑝(𝑥)(Ω), then one has(i)‖𝑢‖_{𝑝(𝑥),Ω}<1(=1;>1)⇔𝜌_{𝑝(𝑥),Ω}(𝑢)<1(=1;>1), (ii)‖𝑢‖_{𝑝(𝑥),Ω}>1⇒‖𝑢‖^𝑝1_{𝑝(𝑥),Ω}≤𝜌_{𝑝(𝑥),Ω}(𝑢)≤‖𝑢‖^𝑝2_{𝑝(𝑥),Ω}, (iii)‖𝑢‖_{𝑝(𝑥),Ω}<1⇒‖𝑢‖^𝑝2_{𝑝(𝑥),Ω}≤𝜌_{𝑝(𝑥),Ω}(𝑢)≤‖𝑢‖^𝑝1_{𝑝(𝑥),Ω},(iv)‖𝑢‖_{𝑝(𝑥),Ω}=𝑎>0⇔𝜌_{𝑝(𝑥),Ω}(𝑢/𝑎)=1.

Proposition 2.3 (see [14, 15]). If 𝑢,𝑢_𝑛∈𝐿^𝑝(𝑥)(Ω), 𝑛=1,2,…, then the following statements are equivalent to each other:(i)lim_𝑛→∞‖𝑢_𝑛−𝑢‖_{𝑝(𝑥),Ω}=0; (ii)lim_𝑛→∞𝜌_{𝑝(𝑥),Ω}(𝑢_𝑛−𝑢)=0; (iii)𝑢_𝑛→𝑢 in measure in Ω and lim_𝑛→∞𝜌_{𝑝(𝑥),Ω}(𝑢_𝑛)=𝜌_{𝑝(𝑥),Ω}(𝑢).

Proposition 2.4 (see [14]). In 𝑊₀^1,𝑝(𝑥)(Ω) the Poincaré inequality holds, that is, there exists a positive constant 𝐶 such that ‖𝑢‖_{𝑝(𝑥),Ω}≤𝐶‖∇𝑢‖_{𝑝(𝑥),Ω},∀𝑢∈𝑊₀^1,𝑝(𝑥)(Ω).(2.8)
Consequently,  ‖𝑢‖_{1,𝑝(𝑥),Ω}=‖|∇𝑢|‖_{𝑝(𝑥),Ω} are equivalent norms on 𝑊₀^1,𝑝(𝑥)(Ω). In what follows, 𝑊₀^1,𝑝(𝑥)(Ω), with 𝑝∈𝐶₊(Ω), will be considered as endowed with the norm ‖𝑢‖_1,𝑝(𝑥). We will use ‖𝑢‖_{1,𝑝(𝑥),Ω}=‖∇𝑢‖_{𝑝(𝑥),Ω} for 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω) in the following discussions.

Proposition 2.5 (see [14, 16]). Assume that Ω is bounded, the boundary of Ω possesses the cone property and 𝑝∈𝐶₊(Ω). If 𝑞∈𝐶₊(Ω) and 𝑞(𝑥)<𝑝^∗(𝑥) for all𝑥∈Ω then the embedding 𝑊^1,𝑝(𝑥)(Ω)↪𝐿^𝑞(𝑥)(Ω) is the compact and continuous.

Lemma 2.6 (see [17]). Assume that Ω is bounded and has a Lipschitz boundary with the cone property and 𝑝∈𝐶₊(Ω). Then there is a continuous boundary trace embedding 𝑊^1,𝑝(⋅)(Ω)↪𝐿^𝑝(⋅)(𝜕Ω).

Lemma 2.7 (see [17]). For any 𝑢∈𝑊^1,𝑝(𝑥)(Ω), let ‖𝑢‖_𝜕∶=‖∇𝑢‖_{𝑝(𝑥),Ω}+‖𝑢‖_{𝑝(𝑥),𝜕Ω}.(2.9) Then, ‖𝑢‖_𝜕 is a norm on 𝑊^1,𝑝(𝑥)(Ω), equivalent to the standard norm of 𝑊^1,𝑝(𝑥)(Ω).

Our analysis is based on the Sobolev space𝐸=(𝑢,𝜐)∈𝑊^1,𝑝(𝑥)Ω₁×𝑊_Γ^1,𝑝(𝑥)Ω₂∶𝑢=𝜐onΣ,(2.10) where𝑊_Γ^1,𝑝(𝑥)Ω₂=𝜐∈𝑊^1,𝑝(𝑥)Ω₂∶𝜐=0onΓ.(2.11) Then, we have following lemma that will permit the variational setting of the problem (P).

Lemma 2.8. 𝐸 is a closed subspace of 𝑊^1,𝑝(𝑥)(Ω₁)×𝑊^1,𝑝(𝑥)(Ω₂), and ‖(𝑢,𝜐)‖_𝐸=‖∇𝑢‖_{𝑝(𝑥),Ω1}+‖∇𝜐‖_{𝑝(𝑥),Ω2}(2.12) defines a norm in 𝐸, equivalent to the standard norm of 𝑊^1,𝑝(𝑥)(Ω₁)×𝑊^1,𝑝(𝑥)(Ω₂).

Proof. It is clear that (2.12) defines a seminorm. Then suppose that ‖(𝑢,𝜐)‖_𝐸=0. Since ‖∇𝜐‖_{𝑝(𝑥),Ω2} defines a norm on 𝑊^1,𝑝(𝑥)(Ω₂) and Poincaré inequality holds in 𝑊^1,𝑝(𝑥)(Ω₂), thus 𝜐=0. From transmission condition, 𝑢=𝜐 on Σ, and therefore 𝑢∈ 𝑊₀^1,𝑝(𝑥)(Ω₁). This shows that (𝑢,𝜐)=0, and hence (2.12) defines a norm in 𝐸. Now, applying the trace theorem in 𝑊^1,𝑝(𝑥)(Ω₂), there exists 𝑐>0 such that for all 𝜐∈𝑊_Γ^1,𝑝(𝑥)(Ω₂)‖𝜐‖_{𝑝(𝑥),Σ}≤𝐶‖∇𝜐‖_{𝑝(𝑥),Ω2}.(2.13) But it is known that ‖∇𝑢‖_{𝑝(𝑥),Ω1}+‖𝑢‖_{𝑝(𝑥),Σ} defines an equivalent norm in 𝑊^1,𝑝(⋅)(Ω₁) from Lemma 2.7. Then combining these two remarks the result follows.

3. Main Results

Let us precisely describe our assumptions in order to establish the main result. The energy functional corresponding to problem (P) is defined as 𝐽∶𝐸→ℝ,𝐽(𝑢,𝜐)=𝑀_Ω1|∇𝑢|^𝑝(𝑥)𝑝(𝑥)𝑑𝑥+𝑁_Ω2|∇𝜐|^𝑝(𝑥)𝑝(𝑥)−𝑑𝑥_Ω1𝐹(𝑥,𝑢)𝑑𝑥−_Ω2𝐺(𝑥,𝜐)𝑑𝑥,(3.1) where 𝑀(𝑡)=∫^𝑡₀𝑀(𝜉)𝑑𝜉, and 𝐹(𝑥,𝑢)=∫^𝑢₀𝑓(𝑥,𝜉)𝑑𝜉. It is not difficult to show that 𝐽∈𝐶¹(𝐸,ℝ), and as a matter of fact, 𝐽 is of class 𝐶¹ and weakly lower semicontinuous. In particular we have, for all (𝑢,𝑣), (𝜑,𝜓)∈𝐸,𝐽^′(𝑢,𝜐),𝜑,𝜓=𝑀_Ω1|∇𝑢|^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω1|∇𝑢|^{𝑝(𝑥)−2}∇𝑢∇𝜑𝑑𝑥+𝑁_Ω2|∇𝜐|^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω2|∇𝜐|^{𝑝(𝑥)−2}∇𝜐∇𝜓𝑑𝑥−_Ω1𝑓(𝑥,𝑢)𝜑𝑑𝑥−_Ω2𝑔(𝑥,𝜐)𝜓𝑑𝑥.(3.2)

We assume the following hypotheses for 𝑀 and 𝑁.

There are positive constants 𝑀₁, 𝑀₂, 𝑁₁, 𝑁₂, and 𝛼 such that(𝐸₁):𝑀₁𝑡^𝛼−1≤𝑀(𝑡)≤𝑀₂𝑡^𝛼−1,(𝐸₂):𝑁₁𝑠^𝛼−1≤𝑁(𝑠)≤𝑁₂𝑠^𝛼−1

for all 𝑡,𝑠>0 and 𝛼≥1.

Now we state our main result.

Theorem 3.1. Let us assume that (𝐸₁) and (𝐸₂) hold. If 𝑞₂<𝛼𝑝₁, then the problem (P) with the transmission condition has at least one nonnegative solution.

Lemma 3.2. There exists 𝜆^∗>0 such that for any 𝜆₁+𝜆₂∈(0,𝜆^∗) there exist 𝜌>0 and 𝑟>0 such that 𝐽_𝜆(𝑢,𝜐)≥𝑟,∀(𝑢,𝜐)∈𝐸,with‖(𝑢,𝜐)‖_𝐸=𝜌.(3.3)

Proof. Since 𝑞(𝑥)<𝑝^∗(𝑥) for all 𝑥∈Ω it follows that (Proposition 2.5) ‖𝑢‖_{𝑞(𝑥),Ω1}≤𝐶₁‖𝑢‖_{1,𝑝(𝑥),Ω1},∀𝑢∈𝑊^1,𝑝(𝑥)Ω₁,‖𝜐‖_{𝑞(𝑥),Ω2}≤𝐶₂‖𝜐‖_{1,𝑝(𝑥),Ω2},∀𝜐∈𝑊^1,𝑝(𝑥)Ω₂.(3.4) By Lemma 2.8 we have ‖𝑢‖_{𝑞(𝑥),Ω1}+‖𝜐‖_{𝑞(𝑥),Ω2}≤𝐶₁‖𝑢‖_{1,𝑝(𝑥),Ω1}+𝐶₂‖𝜐‖_{1,𝑝(𝑥),Ω2}≤𝐶₃‖(𝑢,𝜐)‖_𝐸.(3.5) We fix 𝜌∈(0,1) such that 𝜌<1/𝐶₃. Then the above relation implies ‖𝑢‖_{𝑞(𝑥),Ω1}+‖𝜐‖_{𝑞(𝑥),Ω2}<1,∀(𝑢,𝜐)∈𝐸.(3.6) By Proposition 2.2(ii) and 2.5 and Lemma 2.8, we deduce that _Ω1|𝑢|^𝑞(𝑥)𝑑𝑥+_Ω2|𝑣|^𝑞(𝑥)𝑑𝑥≤‖𝑢‖^𝑞1_{𝑞(𝑥),Ω1}+‖𝑣‖^𝑞1_{𝑞(𝑥),Ω2}≤‖𝑢‖_{𝑞(𝑥),Ω1}+‖𝑣‖_{𝑞(𝑥),Ω2}≤𝐶₄‖𝑢‖_{1,𝑝(𝑥),Ω1}+‖𝑣‖_{1,𝑝(𝑥),Ω2}(3.7)≤𝐶₅‖(𝑢,𝑣)‖_𝐸.(3.8) Using (𝐸₁), (𝐸₂), Proposition 2.2(iii), and (3.8), we obtain that for any (𝑢,𝜐)∈𝐸 with ‖(𝑢,𝜐)‖_𝐸<1 the following inequalities hold true: 𝐽_𝜆(𝑢,𝜐)=𝑀_Ω1|∇𝑢|^𝑝(𝑥)𝑝(𝑥)𝑑𝑥+𝑁_Ω2|∇𝜐|^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω2|∇𝜐|^𝑝(𝑥)𝑑𝑥−𝜆_1Ω1|𝑢|^𝑞(𝑥)𝑞(𝑥)𝑑𝑥−𝜆_2Ω2|𝜐|^𝑞(𝑥)𝑞(𝑥)𝑑𝑥≥𝑀₁^{(1/𝑝2)𝜌1𝑝(𝑥),Ω}₀^(∇𝑢)𝑡^𝛼−1𝑑𝑡+𝑁₁^{(1/𝑝2)𝜌2𝑝(𝑥),Ω}₀^(∇𝑢)𝜂^𝛼−1−𝜆𝑑𝜂₁𝑞_1Ω1|𝑢|^𝑞(𝑥)𝜆𝑑𝑥−²𝑞_1Ω2|𝜐|^𝑞(𝑥)≥𝑀𝑑𝑥₁𝛼1𝑝_2Ω1|∇𝑢|^𝑝(𝑥)𝑑𝑥^𝛼+𝑁₁𝛼1𝑝_2Ω2|∇𝜐|^𝑝(𝑥)𝑑𝑥^𝛼−𝜆₁𝑞_1Ω1|𝑢|^𝑞(𝑥)𝜆𝑑𝑥−²𝑞_1Ω2|𝜐|^𝑞(𝑥)≥𝑀𝑑𝑥min₁,𝑁₁𝛼𝑝₂^𝛼‖∇𝑢‖^𝛼𝑝2_{𝑝(𝑥),Ω1}+‖∇𝜐‖^𝛼𝑝2_{𝑝(𝑥),Ω2}−𝐶₅𝜆₁+𝜆₂𝑞₁‖(𝑢,𝜐)‖_𝐸≥2^{1−𝛼𝑝2}min𝑀₁,𝑁₁𝛼𝑝₂^𝛼‖∇𝑢‖_{𝑝(𝑥),Ω1}+‖∇𝜐‖_{𝑝(𝑥),Ω2}^𝛼𝑝2−𝐶₅𝜆₁+𝜆₂𝑞₁‖(𝑢,𝜐)‖_𝐸≥2^{1−𝛼𝑝2}min𝑀₁,𝑁₁𝛼𝑝₂^𝛼‖(𝑢,𝜐)‖^𝛼𝑝2_𝐸−𝐶₅𝜆₁+𝜆₂𝑞₁‖(𝑢,𝜐)‖_𝐸.(3.9) By the above inequality if we define 𝜆^∗=2^{1−𝛼𝑝2}𝜌^{𝛼𝑝2−1}𝑞₁min𝑀₁,𝑁₁𝐶₅𝛼𝑝₂^𝛼,(3.10) then for any 𝜆₁+𝜆₂∈ (0,𝜆^∗) and (𝑢,𝜐)∈𝐸 there exists 𝑟>0 such that 𝐽_𝜆(𝑢,𝜐)≥𝑟>0.
The proof of Lemma 3.2 is complete.

Lemma 3.3. There exists 𝜑,𝜙∈𝐸 such that 𝜑,𝜙≥0 and 𝜑,𝜙≠0 and 𝐽_𝜆(𝑡𝜑,𝑡𝜙)<0 for 𝑡>0 small enough.

Proof. Let |Ω₁|>0 and |Ω₂|>0. Moreover, let choose 𝜑,𝜙∈𝐶^∞₀(Ω) and 0≤𝜑(𝑥)≤1 in Ω₁,  0≤𝜙(𝑥)≤1 in Ω₂. Then, for any 𝑡∈(0,1), by (𝐸₁) and (𝐸₂) it follows 𝐽_𝜆𝑡𝜑,𝑡𝜙=𝑀_Ω1𝑡∇𝜑^𝑝(𝑥)𝑝(𝑥)𝑑𝑥+𝑁_Ω2𝑡∇𝜙^𝑝(𝑥)𝑝(𝑥)𝑑𝑥−𝜆_1Ω1𝑡𝜑^𝑞(𝑥)𝑞(𝑥)𝑑𝑥−𝜆_2Ω2𝑡𝜙^𝑞(𝑥)𝑞(𝑥)𝑑𝑥≤𝑀₂^{(1/𝑝1)𝜌1𝑝(𝑥),Ω}₀^{(𝑡∇𝜑)}𝑠^𝛼−1𝑑𝑠+𝑁₂^{(1/𝑝1)𝜌2𝑝(𝑥),Ω}₀^{(𝑡∇𝜙)}𝜂^𝛼−1𝑑𝜂−𝜆_1Ω1𝑡𝜑^𝑞(𝑥)𝑞(𝑥)𝑑𝑥−𝜆_2Ω2𝑡𝜙^𝑞(𝑥)𝑞(𝑥)≤𝑀𝑑𝑥₂𝑡^𝛼𝑝1𝛼𝑝₁^𝛼𝜌^𝛼_{𝑝(𝑥),Ω1}+𝑁∇𝜑²𝑡^𝛼𝑝1𝛼𝑝₁^𝛼𝜌^𝛼_{𝑝(𝑥),Ω2}−𝜆∇𝜙₁𝑡^𝑞2𝑞₂𝜌_{𝑞(𝑥),Ω1}𝜑−𝜆₂𝑡^𝑞2𝑞₂𝜌_{𝑞(𝑥),Ω2}𝜙.(3.11) Let 𝑅_𝛼=𝑀₂𝜌^𝛼_{𝑝(𝑥),Ω1}∇𝜑+𝑁₂𝜌^𝛼_{𝑝(𝑥),Ω2},𝑆∇𝜙_𝜆1,𝜆2=𝜆₁𝜌_{𝑞(𝑥),Ω1}𝜑+𝜆₂𝜌_{𝑞(𝑥),Ω2}𝜙.(3.12) Then 𝐽_𝜆≤𝑡𝑡𝜑,𝑡𝜙^𝛼𝑝1𝛼𝑝₁^𝛼𝑅_𝛼−𝑡^𝑞2𝑞₂𝑆_𝜆1,𝜆2.(3.13) Therefore, we conclude 𝐽_𝜆𝑡𝜑,𝑡𝜙<0,(3.14) for 0<𝑡<𝜎^{1/𝛼𝑝1−𝑞2} providing that 𝛼𝑝0<𝜎<𝑚𝑖𝑛1,^1𝛼𝑆_𝜆1,𝜆2𝑞₂𝑅_𝛼.(3.15) The proof of Lemma 3.3 is complete.

Proof of Theorem 3.1. From Lemma 3.3, we infer that there exists a ball centered at the origin 𝐵_𝜌(0)⊂𝐸, such that inf_{𝜕𝐵𝜌(0)}𝐽_𝜆>0.(3.16) Furthermore, by Lemma 3.3, we know that there exists (𝜑,𝜔)∈𝐸 such that 𝐽_𝜆(𝑡𝜑,𝑡𝜔)<0 for 𝑡>0 small enough. Therefore, considering also inequality (3.14), we obtain that −∞<𝑐∶=inf_𝐵𝜌(0)𝐽_𝜆<0.(3.17) Let us choose 𝜀>0. Then, it follows that 0<𝜀≤inf_{𝜕𝐵𝜌(0)}𝐽_𝜆−inf_𝐵𝜌(0)𝐽_𝜆.(3.18) Now, if we apply the Ekeland's variational principle [18] to the functional 𝐽_𝜆∶𝐵_𝜌(0)→ℝ, it follows that there exists (𝑢_𝜀,𝜐_𝜀)∈𝐵_𝜌(0) such that 𝐽_𝜆𝑢_𝜀,𝜐_𝜀<inf_𝐵𝜌(0)𝐽_𝜆𝐽+𝜀,_𝜆𝑢_𝜀,𝜐_𝜀<𝐽_𝜆(𝑢,𝜐)+𝜀𝑢−𝑢_𝜀,𝜐−𝜐_𝜀𝐸,𝑢_𝜀≠𝑢,𝜐≠𝜐_𝜀.(3.19) By the fact that 𝐽_𝜆𝑢_𝜀,𝜐_𝜀<inf_𝐵𝜌(0)𝐽_𝜆+𝜀<inf_𝐵𝜌(0)𝐽_𝜆+𝜀<inf_{𝜕𝐵𝜌(0)}𝐽_𝜆,(3.20) we can infer that (𝑢_𝜀,𝜐_𝜀)∈𝐵_𝜌(0).
Now, let us define Φ_𝜆∶𝐵_𝜌(0)→ℝ by Φ_𝜆(𝑢,𝜐)=𝐽_𝜆(𝑢,𝜐)+𝜀‖(𝑢−𝑢_𝜀,𝜐−𝜐_𝜀)‖_𝐸. It is not difficult to see that (𝑢_𝜀,𝜐_𝜀) is a minimum point of Φ_𝜆, and thus Φ_𝜆𝑢_𝜀+𝑡⋅𝜏,𝜐_𝜀+𝑡⋅𝜎−Φ_𝜆𝑢_𝜀,𝜐_𝜀𝑡≥0,(3.21) for 𝑡>0 small enough and any (𝜏,𝜎)∈𝐵₁(0). By the above expression, we have 𝐽_𝜆𝑢_𝜀+𝑡⋅𝜏,𝜐_𝜀+𝑡⋅𝜎−𝐽_𝜆𝑢_𝜀,𝜐_𝜀𝑡+𝜀‖(𝜏,𝜎)‖_𝐸≥0.(3.22) Letting 𝑡→0, we have ⟨𝐽^′_𝜆𝑢_𝜀,𝜐_𝜀,(𝜏,𝜎)⟩+𝜀‖(𝜏,𝜎)‖_𝐸>0,(3.23) and this implies that ‖𝐽^′_𝜆(𝑢_𝜀,𝜐_𝜀)‖≤𝜀. So, we infer that there exists a sequence {𝜔_𝑛,𝜈_𝑛}⊂𝐵_𝜌(0) such that 𝐽_𝜆𝜔_𝑛,𝜈_𝑛⟶𝑐=inf_𝐵𝜌(0)𝐽_𝜆<0,𝐽^′_𝜆𝜔_𝑛,𝜈_𝑛⟶0.(3.24)
It is obvious that {𝜔_𝑛,𝜈_𝑛} is bounded in 𝐸. Therefore, there exists (𝜔,𝜈)∈𝐸 such that, up to a subsequence, {𝜔_𝑛,𝜈_𝑛} converges weakly to (𝜔,𝜈) in 𝐸 and converges strongly to 𝜔_𝑛→𝜔 in 𝐿^𝑞(𝑥)(Ω₁) and 𝜈_𝑛→𝜈 in 𝐿^𝑞(𝑥)(Ω₂) (Proposition 2.5). Thus ⟨𝐽^′_𝜆(𝑢_𝑛,𝜐_𝑛),(𝑢_𝑛−𝑢,𝜐_𝑛−𝜐)⟩→0 and ⟨𝐽^′_𝜆𝜔_𝑛,𝜈_𝑛,𝜔_𝑛−𝜔,𝜈_𝑛−𝜈⟩=𝑀_Ω1∇𝜔_𝑛^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω1∇𝜔_𝑛^{𝑝(𝑥)−2}∇𝜔_𝑛∇𝜔_𝑛−∇𝜔𝑑𝑥+𝑁_Ω2∇𝜈_𝑛^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω2∇𝜈_𝑛^{𝑝(𝑥)−2}∇𝜈_𝑛∇𝜈_𝑛−∇𝜈𝑑𝑥−𝜆_1Ω1𝜔_𝑛^{𝑞(𝑥)−2}𝜔_𝑛𝜔_𝑛−𝜔𝑑𝑥−𝜆_2Ω2𝜈_𝑛^{𝑞(𝑥)−2}𝜈_𝑛𝜈_𝑛−𝜈𝑑𝑥⟶0.(3.25) By Proposition 2.1, it follows that _Ω1𝜔_𝑛^{𝑞(𝑥)−2}𝜔_𝑛𝜔_𝑛−𝜔𝑑𝑥≤𝜔_𝑛^{𝑞(𝑥)−1}_{𝑞′(𝑥),Ω1}𝜔_𝑛−𝜔_{𝑞(𝑥),Ω1},_Ω2𝜈_𝑛^{𝑞(𝑥)−2}𝜈_𝑛𝜈_𝑛−𝜈𝑑𝑥≤𝜈_𝑛^{𝑞(𝑥)−1}_{𝑞′(𝑥),Ω2}𝜈_𝑛−𝜈_{𝑞(𝑥),Ω2}.(3.26) Since {𝜔_𝑛} converges strongly to 𝜔 in 𝐿^𝑞(𝑥)(Ω₁), that is, ‖𝜔_𝑛−𝜔‖_{𝑞(𝑥),Ω1}→0 as 𝑛→∞, we get _Ω1𝜔_𝑛^{𝑞(𝑥)−2}𝜔_𝑛𝜔_𝑛−𝜔𝑑𝑥⟶0(3.27) and similarly _Ω2𝑣_𝑛^{𝑞(𝑥)−2}𝜈_𝑛𝜈_𝑛−𝜈𝑑𝑥⟶0.(3.28) Hence, 𝑀_Ω1∇𝜔_𝑛^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω1∇𝜔_𝑛^{𝑝(𝑥)−2}∇𝜔_𝑛∇𝜔_𝑛𝑁−∇𝜔𝑑𝑥⟶0,_Ω2∇𝜈_𝑛^𝑝(𝑥)𝑝(𝑥)𝑑𝑥_Ω2∇𝜈_𝑛^{𝑝(𝑥)−2}∇𝜈_𝑛∇𝜈_𝑛−∇𝜈𝑑𝑥⟶0.(3.29) From (𝐸₁) and (𝐸₂), it follows that _Ω1∇𝜔_𝑛^{𝑝(𝑥)−2}∇𝜔_𝑛∇𝜔_𝑛−∇𝜔𝑑𝑥⟶0,_Ω2∇𝜈_𝑛^{𝑝(𝑥)−2}∇𝜈_𝑛∇𝜈_𝑛−∇𝜈𝑑𝑥⟶0.(3.30) Eventually, by [19, Theorem  3.1], we get that {𝜔_𝑛,𝜈_𝑛} converges strongly to (𝜔,𝜈) in 𝐸, so we conclude that (𝜔,𝜈) is a nontrivial weak solution for problem (P). The proof of Theorem 3.1 is complete.

Acknowledgments

The authors would like to thank the referees for their many helpful suggestions and corrections, which improve the presentation of this paper. This research was supported by DUBAP grant no. 10-FF-15, Dicle University, Turkey.