On a nonlocal elliptic system with transmission conditions

1 Introduction

The interest towards partial differential equations with nonstandard growth conditions is due to applications. It arises from the study of electrorheological fluids, nonlinear elasticity problems, image processing, mathematical description of the processes filtration of an ideal barotropic gas through a porous medium, etc.; see, for example, [2, 7, 11, 20] and references therein.

The study of transmission problems is a new and interesting topic that arises 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 in solid mechanics too, if a body consists of composite materials. We refer the reader to [19] for nonlinear elliptic transmission problems and to [13] for a nonlinear nonlocal elliptic transmission problem. Furthermore, the uniqueness and regularity of the solutions to the thermoelastic transmission problem were investigated in [14].

The main aim of this paper is to study the existence of nontrivial weak solution for nonlocal elliptic systems of gradient type with nonstandard growth conditions. More precisely, let Ω be a smooth bounded domain of ℝN, N≥2, and let Ω1⊂Ω be a subdomain with smooth boundary Σ satisfying Ω¯1⊂Ω. Writing Γ=∂⁡Ω and Ω2=Ω∖Ω¯1, we have Ω=Ω¯1∪Ω2 and ∂⁡Ω2=Σ∪Γ, and that η is the outward normal to Ω2 and is inward to Ω1. We consider the following system:

with the transmission conditions

where p∈C⁢(Ω¯), η is outward normal to Ω2 and is inward to Ω1. The functions M1 and M2:ℝ+→ℝ are continuous, and the following conditions hold:

1. The function a:Ω¯1→ℝ belongs to Lα0⁢(x)⁢(Ω1), with α0∈C+⁢(Ω¯1), and we have

   N⁢p⁢(x)N⁢p⁢(x)-α⁢(x)⁢(N-p⁢(x))<α0⁢(x)<p⁢(x)p⁢(x)-α⁢(x) for all ⁢x∈Ω¯1.

2. The function b:Ω¯2→ℝ+ is nonnegative and belongs to b∈Lr⁢(x)⁢(Ω2), with r⁢(x)∈C+⁢(Ω¯2), and we have

   1<β⁢(x)<r⁢(x)-1r⁢(x)⁢p*⁢(x),

   where p*⁢(x)=n⁢p⁢(x)n-p⁢(x) with p+:=ess⁢supx∈Ω¯⁡p⁢(x)<N,

   C+⁢(Ω¯):={h:h∈C⁢(Ω¯)⁢ and ⁢h⁢(x)>1⁢ for all ⁢x∈Ω¯},

   and h+:=maxΩ¯⁡h⁢(x), h-:=minΩ¯⁡h⁢(x) for any h∈C+⁢(Ω¯).

We point out that (P) with the transmission condition is a generalization of the stationary problem of two wave equations of Kirchhoff type,

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 [16, 18].

Problems like this are called Kirchhoff type problems. In recent years, many interesting results for problems of Kirchhoff type were obtained. We refer the reader to [1, 3, 4, 10, 17, 6, 5] and references therein for details.

In [4], the Cekic and Mashiyev proved the existence of nontrivial solution for (P) in the case a⁢(x)=b⁢(x)=1 and α⁢(x)=β⁢(x)=q⁢(x), when q∈C⁢(Ω¯) and 1<q⁢(x)<p*⁢(x) for all x∈Ω¯, where p*⁢(x)=N⁢p⁢(x)N-p⁢(x) if p⁢(x)<N or p*⁢(x)=∞ otherwise.

More recently, in [3], using the mountain pass theorem combined with Ekeland’s variational principle, Ayoujil and Moussaoui proved the existence of at least two distinct nontrivial weak solutions for (P), which generalizes the work [4]. However, nonlocal transmission problems with variable exponents with singular coefficients are rare.

Inspired by the above papers, our goal is to present a result of a weak solution to a kind of transmission problem with singular coefficients. Using the mountain pass theorem, we establish that problem (P) has at least one nontrivial weak solution under appropriate conditions on the nonlinear perturbations f and g.

We confine ourselves to the case where M1=M2 for simplicity. Notice that the results of this paper remain valid for M1≠M2 by adding some slight changes in the hypothesis (M1) and (M2) below.

Throughout this paper, we will assume that M:ℝ+→ℝ+ is a continuous function satisfying the following assumptions:

1. There exist real numbers m1,m2>0 and θ2>θ1>1 such that

   m1⁢tθ1-1≤M⁢(t)≤m2⁢tθ2-1 for all ⁢t>0.

2. There exists 0<ν<1 such that

   M^⁢(t)≥(1-ν)⁢M⁢(t)⁢t for all t∈ℝ+,

   where M^⁢(t)=∫0tM⁢(s)⁢𝑑s.

We suppose also that α,β∈C+⁢(Ω¯) such that

Now, we are ready to state our main result.

Here, any solution of problem (P) with the transmission conditions will belong to the framework of the generalized Sobolev space, which will be briefly described in the following section, that is,

where

The paper consists of three sections. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we give the proof of our main result.

2 Preliminary results and notations

In order to guarantee the integrity of the paper, we recall some definitions and basic properties of variable exponent Lebesgue–Sobolev spaces. We refer to [9, 12] for the fundamental properties of these spaces, and for further details.

Let σ:Ω→ℝ be a measurable real function such that σ⁢(x)>0 for a.e. x∈Ω. For p∈C+⁢(Ω), define the weighted variable exponent Lebesgue space

Equipped with the so-called Luxemburg norm

Lσ⁢(x)p⁢(x)⁢(Ω) becomes a separable reflexive Banach space. In particular, when σ⁢(x)≡1 on Ω, Lσ⁢(x)p⁢(x)⁢(Ω) is the usual variable exponent Lebesgue space Lp⁢(x)⁢(Ω).

The mapping ρp⁢(x),σ⁢(x),Ω:Lσ⁢(x)p⁢(x)⁢(Ω)→ℝ, defined by

is called the modular of the Lσ⁢(x)p⁢(x)⁢(Ω) space and plays an important role in manipulating the generalized Lebesgue spaces. The following proposition illuminates the close relation between the |u|p⁢(x),σ⁢(x),Ω and the convex modular ρp⁢(x),σ⁢(x),Ω.

As in the constant exponent case, for any positive integer k, set

Endowed with the norm

the space Wk,p⁢(x)⁢(Ω) becomes a separable reflexive Banach space. In W01,p⁢(x)⁢(Ω), which denotes the closure of C0∞⁢(Ω) in Wk,p⁢(x)⁢(Ω), the Poincaré inequality holds, that is, there exists a positive constant C such that

So, |∇⁡u|p⁢(x),Ω is an equivalent norm in W01,p⁢(x)⁢(Ω). We will use the equivalent norm in the following discusion and write ∥u∥p⁢(x),Ω=|∇⁡u|p⁢(x),Ω for simplicity.

Recall the following lemma, see [4, Lemma 2.8], which will permit the variational setting of problem (P).

3 Proof of the main result

For simplicity, we use the letters c, ci,i=1,2,…, to denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. Note that 〈⋅,⋅〉 denotes the dual pair.

The energy functional J:E→ℝ associated to problem (P) is defined as

where

and

In a standard way, it can be proved that J∈C1⁢(E,ℝ). Moreover, for all (φ,ψ)∈E, we have

where

and

Thus, the weak solutions of problem (P) are exactly the critical points of the functional J. Due to conditions (M1) and (1.1), we can show that J is weakly lower semi-continuous in E.

In order to establish Theorem 1.1, we need the following lemmas which play an important role in our arguments.