Multiple Solutions for Elliptic -Kirchhoff-Type Potential Systems in Unbounded Domains

Abstract

In this paper, we establish the existence of at least three weak solutions for a parametric double eigenvalue quasi-linear elliptic -Kirchhoff-type potential system. Our approach is based on a variational method, and a three critical point theorem is obtained by Bonano and Marano.

1. Introduction

The aim of this paper is to show the existence of at least three weak solutions for the following class of nonlocal quasi-linear elliptic systems in :where , and , is a positive real parameter, and , such that and . and are bounded continuous functions, belongs to and satisfies adequate growth assumptions, and denotes the partial derivative of with respect to (respectively, ). Here, we denote the so-called -Laplacian operator, and for ,

System (1) is a generalization of the elliptic equation associated with the following Kirchhoff equation, introduced by Kirchhoff in [1]:where , , , and are constants. This equation extends classical D’Alembert’s wave equation by considering the effects of the changes on the length of the strings during the vibrations. A distinguishing feature of equation (3) is that the equation contains a nonlocal coefficient which depends on the average , and hence, the equation is no longer a pointwise equation. The parameters in equation (3) have the following meanings: is Young’s modulus of the material, is the mass density, is the length of the string, is the area of cross section, and is the initial tension.

The -Laplacian operator possesses more complicated nonlinearities than -Laplacian operator mainly due to the fact that it is not homogeneous. The study of various mathematical problems involving variable exponents has received a strong rise of interest in recent years. We can, for example, refer to [2–12]. This great interest may be justified by their various physical applications. In fact, there are applications concerning nonlinear elasticity theory [13], electrorheological fluids [14, 15], stationary thermorheological viscous flows [16], and continuum mechanics [17]. It also has wide applications in different research fields, such as image processing model [18] and the mathematical description of the filtration process of an ideal barotropic gas through a porous medium [19].

The existence and multiplicity of solutions for the elliptic systems involving the -Kirchhoff model have been studied by many authors, where the nonlinear source has different mixed growth conditions. We refer the reader to see [20–22] and the references therein for an overview on this subject. In connection to our context, the author obtained in [23] the existence and multiplicity of solutions for the vector-valued elliptic system:where is a bounded domain in , with smooth boundary , and , and and are continuous functions such that . The author applies a direct variational approach and the theory of variable exponent Sobolev spaces.

On the contrary, by using the mountain pass theorem, the authors in [24] showed the existence of nontrivial solutions for system (1) when , and are continuous functions such that , , , and verifies some mixed growth conditions.

The goal of this work is to establish the existence of a definite interval in which lies such that system (1) admits at least three weak solutions by applying the following very recent abstract critical point result of Bonanno and Marano [25], which is a more precise version of Theorem 3.2 of [26].

Lemma 1 (see [25], Theorem 3.6). Let be a reflexive real Banach space; be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on ; and be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such thatAssume that there exist and , with , such that(a₁)(a₂)For each , the functional is coerciveThen, for each , the functional has at least three distinct critical points in .

The rest of the paper is organized as follows. Section 2 contains some basic preliminary knowledge of the variable exponent spaces and some results that we shall use here. Finally, in Section 3, we state and establish our main result.

2. Preliminaries and Basic Notations

First, we introduce the definitions of Lebesgue–Sobolev spaces with variable exponents. The details can be found in [27–29]. Denote as the set of all measurable real functions on . Set

For any , we define

For any , we define the variable exponent Lebesgue space asendowed with the Luxemburg norm

Let be such that , for a.e . Define the weighted variable exponent Lebesgue space :with the norm

From now on, we suppose that with . Then, obviously, is a Banach space (see [30] for details).

On the contrary, the variable exponent Sobolev space is defined as follows:and is endowed with the norm

Next, the weighted variable exponent Sobolev space is defined asand is endowed with the norm

Note that and are equivalent norms in . Moreover, when , it is well known that , , and are separable, reflexive, and uniformly convex Banach spaces.

Now, we display some facts that we shall use later.

Proposition 1 (see [27, 28]). The conjugate space of is , whereMoreover, for any , we have

Proposition 2 (see [27, 28]). Denote , for all . We haveand the following implications are true:(i) (ii)(iii)

Denote , for all . From Proposition 2, we have

Proposition 3 (see [31]). Let and be measurable functions such that and almost everywhere in . If , , then we haveIn particular, if is constant, then

For all , denotethe critical Sobolev exponent of .

Proposition 4 (see [27, 31]). Let , the space of Lipschitz-continuous functions defined on . There exists a positive constant such that

Proposition 5 (see [27, 31]). Assume that satisfies for each . If is such that , for each , then there exists a continuous and compact embedding .

In the following, we shall use the product spaceequipped with the normwhere (respectively, ) is the norm in (respectively, ) defined above. We denote as the dual space of equipped with the usual dual norm.

Definition 1. is called a weak solution of system (1) iffor all , where is defined in (2).
We denote as the energy functional associated with problem (1):where are defined as follows:wherefor any in , withNote that we have the following formula:It is well know that and that critical points of correspond to weak solutions of problem (1).

2.1. Hypotheses

In this paper, we use the following assumptions:(H1) and .(H2)There exist positive functions and such that where and , , for all , and the weight functions (respectively, ) belong to the generalized Lebesgue spaces (respectively, ), with(H3) are continuous and increasing functions such that , for all , .(H4)There exist and such that the following conditions are satisfied:(C1),(C2),wherewith and and representing the constants defined in Proposition 4.

3. The Main Results

We will use the three critical point theorem obtained by Bonano and Marano together with the following lemmas to get our main results.

Lemma 2. The functional is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous, coercive whose Gâteaux derivative admits a continuous inverse on .

Proof. It is well known that the functional is well defined and is continuously Gâteaux differentiable functional whose derivative at the point is the functional given bywherefor every and is defined in (19).
Let us show that is coercive. By using (19) and (20), we have for all ,This shows that as , that is, is coercive on .
Now, in order to show that the operator is strictly monotone, it suffices to prove that is strictly convex.
For , the functional defined in (2) is clearly a Gâteaux derivative at any , and his derivative is given byfor all .
Taking into account the inequality (see, e.g., Chapter I in [32]) for , there exists a positive constant such thatfor any . Therefore, we havefor all which means that is strictly monotone. So, by ([33], Proposition 25.10), is strictly convex. Moreover, since the Kirchhoff function is nondecreasing, is convex in . Thus, for every with and every with , one hasThis shows that is strictly convex in . Similarly, we have that is strictly convex in . Hence, is strictly convex in X, and so is strictly monotone.
It is clear that is an injection since is a strictly monotone operator in X. Moreover, since we havethen, we deduce that is coercive (see (19)). Thus, is a surjection. Now, since is hemicontinuous in , then by applying (Proposition 4.2, [22]), we conclude that admits a continuous inverse on . Moreover, the monotonicity of on ensures that is sequentially lower semicontinuously on (see [33], Proposition 25. 20). The proof of the lemma is complete.

Lemma 3 (see [8]). Under assumptions and , the functional is well defined and is of class on . Moreover, its derivative is given byMoreover, is compact from to .

Theorem 1. Under assumptions , system (1) admits at least three distinct weak solutions in for each

Proof. By Lemma 2, is coercive, and by the definitions of and and from hypothesis , we have . Moreover, the required hypothesis follows from condition and the definition of by choosing . On the contrary, by applying Proposition 4 for , we havewith , defined in (36). Now, from (47), we obtain for where is defined in (35). Then,Therefore, from condition , we havefrom which condition of Lemma 1 follows.
To show that the functional is coercive, we use inequality (3.8). For all , we have in virtue of and Using Young’s inequality, we obtainThis shows that as since we have , that is, is coercive on , for every parameter , in particular, for every . Then, condition in Lemma 1 also holds. Now, all the hypotheses of Lemma 1 are satisfied. Note that the solutions of the equation are exactly the weak solutions of (1). Thus, for eachsystem (1) admits at least three weak solutions in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.