Multiple Solutions for Elliptic (p(x), q(x))-Kirchhoff-Type Potential Systems in Unbounded Domains

where N≥ 2, p and q ∈ C∗(RN) ≔ r ∈ C(R { ): 1< r � infx∈RN r(x)< r(x)< r � supx∈RN r(x)<N,∀x ∈ R }, λ is a positive real parameter, and a, b ∈ L∞(RN) such that a ≔ ess infx∈RN a(x)> 0 and b ≔ ess infx∈RN b(x)> 0. M1 and M2 are bounded continuous functions, F belongs to C1(RN × R2) and satisfies adequate growth assumptions, and Fu(respectively, Fv) denotes the partial derivative of F with respect to u (respectively, v). Here, we denote Δp(x)u ≔ div(|∇u| p(x)− 2∇u) the so-called p(x)-Laplacian operator, and for u ∈ C∗(RN),


Introduction
e aim of this paper is to show the existence of at least three weak solutions for the following class of nonlocal quasilinear elliptic systems in R N : where N ≥ 2, p and q ∈ C * (R N ) ≔ r ∈ C(R { N ): 1 < r − � inf x∈R N r(x) < r(x) < r + � sup x∈R N r(x) < N, ∀x ∈ R N }, λ is a positive real parameter, and a, b ∈ L ∞ (R N ) such that a ≔ ess inf x∈R N a(x) > 0 and b ≔ ess inf x∈R N b(x) > 0. M 1 and M 2 are bounded continuous functions, F belongs to C 1 (R N × R 2 ) and satisfies adequate growth assumptions, and F u (respectively, F v ) denotes the partial derivative of F with respect to u (respectively, v). Here, we denote Δ p(x) u ≔ div(|∇u| p(x)− 2 ∇u) the so-called p(x)-Laplacian operator, and for u ∈ C * (R N ), |∇|u r(x) + a(x)|u| r(x) dx.
System (1) is a generalization of the elliptic equation associated with the following Kirchhoff equation, introduced by Kirchhoff in [1]: where ρ, ρ 0 , E, and L are constants. is 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 (ρ 0 /h)+ (E/2L) L 0 |zu/zx| 2 dx which depends on the average (1/2L) L 0 |zu/zx| 2 dx, and hence, the equation is no longer a pointwise equation. e parameters in equation (3) have the following meanings: E is Young's modulus of the material, ρ is the mass density, L is the length of the string, h is the area of cross section, and ρ 0 is the initial tension. e p(x)-Laplacian operator possesses more complicated nonlinearities than p-Laplacian operator mainly due to the fact that it is not homogeneous. e 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][3][4][5][6][7][8][9][10][11][12]. is 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]. e existence and multiplicity of solutions for the elliptic systems involving the p(x)-Kirchhoff model have been studied by many authors, where the nonlinear source F has different mixed growth conditions. We refer the reader to see [20][21][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 Ω is a bounded domain in R N , with smooth boundary zΩ, and M 2 (t) are continuous functions such that M 1 (t) � M 2 (t). e 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 (p, q) ∈ [C(R N )] 2 (N ≥ 2), M 1 (t) and M 2 (t) are continuous functions such that M 1 (t) � M 2 (t), a(x) � b(x) � 0, λ � 1, and F ∈ C 1 (R N × R 2 , R) verifies some mixed growth conditions. e goal of this work is to establish the existence of a definite interval in which λ 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 eorem 3.2 of [26].
Lemma 1 (see [25], eorem 3.6). Let X be a reflexive real Banach space; Φ: X ⟶ R be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X * ; and Ψ: X ⟶ R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist e > 0 and x ∈ X, with e < Φ(x), such that en, for each λ ∈ Λ e , the functional Φ − λΨ has at least three distinct critical points in X. e 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.

Preliminaries and Basic Notations
First, we introduce the definitions of Lebesgue-Sobolev spaces with variable exponents. e details can be found in [27][28][29]. Denote M(R N ) as the set of all measurable real functions on R N . Set For any p ∈ C + (R N ), we define For any p ∈ C + (R N ), we define the variable exponent Lebesgue space as endowed with the Luxemburg norm with the norm From now on, we suppose that a ∈ L ∞ (R N ) with a ≔ ess inf x∈R N a(x) > 0. en, obviously, L p(x) a is a Banach space (see [30] for details).
On the contrary, the variable exponent Sobolev space W 1,p(x) (R N ) is defined as follows: (12) and is endowed with the norm Next, the weighted variable exponent Sobolev space W (14) and is endowed with the norm Note that ‖ · ‖ a and ‖ · ‖ 1,p(x) are equivalent norms in W reflexive, and uniformly convex Banach spaces. Now, we display some facts that we shall use later.
Proposition 4 (see [27,31]). Let p ∈ C 0,1 + (R N ), the space of Lipschitz-continuous functions defined on R N . ere exists a positive constant c such that Proposition 5 (see [27,31]). Assume that p ∈ C(R N ) sat- for each x ∈ R N , then there exists a continuous and compact embedding In the following, we shall use the product space equipped with the norm where ‖ · ‖ a (respectively, (R N )) defined above. We denote X * as the dual space of X equipped with the usual dual norm.
We denote E λ as the energy functional associated with problem (1): where Φ, Ψ: X ⟶ R are defined as follows: where for any w � (u, v) in X, with Note that we have the following formula: It is well know that E λ ∈ C 1 (X, R) and that critical points of E λ correspond to weak solutions of problem (1).

Hypotheses.
In this paper, we use the following assumptions: where 1 < μ 1 , μ 2 , ] 1 , ] 2 < inf(p(x), q(x)) and p(x), q(x) > N/2, for all x ∈ R N , and the weight functions a 1 , b 2 (respectively, a 2 , b 1 ) belong to the generalized Lebesgue spaces L α i (R N ) (respectively, L β (R N )), with .  > 0 and (w 1 , w 2 ) ∈ X such that the following conditions are satisfied: with t > 0 and c p(x) and c q(x) representing the constants defined in Proposition 4.

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.
e functional Φ is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous, coercive whose Gâteaux derivative admits a continuous inverse on X * . Proof. It is well known that the functional Φ is well defined and is continuously Gâteaux differentiable functional whose derivative at the point (u, v) ∈ X is the functional Φ ′ (u, v) given by where 4 International Journal of Differential Equations for every (φ, ψ) ∈ X and L r (u) is defined in (19).
Let us show that Φ is coercive. By using (19) and (20), we have for all (u, v) ∈ X, Now, in order to show that the operator Φ ′ : X ⟶ X * is strictly monotone, it suffices to prove that Φ is strictly convex.
For r ∈ C * (R N ), the functional L r : (2) is clearly a Gâteaux derivative at any u ∈ W 1,r(x) a (R N ), and his derivative is given by for all φ ∈ W 1,r(x) a (R N ). Taking into account the inequality (see, e.g., Chapter I in [32]) for c > 1, there exists a positive constant C c such that for any α, β ∈ R N . erefore, we have for all u 1 ≠ u 2 ∈ W 1,p(x) a (R N ) which means that L p ′ is strictly monotone. So, by ([33], Proposition 25.10), L p is strictly convex. Moreover, since the Kirchhoff function M 1 is nondecreasing, M 1 is convex in [0, +∞[. us, for every u 1 , u 2 ∈ W 1,p(x) a (R N ) with u 1 ≠ u 2 and every ]s, t ∈ 0, 1[ with s + t � 1, one has Hence, Φ is strictly convex in X, and so Φ ′ � Φ 1 ′ + Φ 2 ′ is strictly monotone.
It is clear that Φ ′ is an injection since Φ ′ is a strictly monotone operator in X. Moreover, since we have then, we deduce that Φ ′ is coercive (see (19)). us, Φ ′ is a surjection. Now, since Φ ′ is hemicontinuous in X, then by applying (Proposition 4.2, [22]), we conclude that Φ ′ admits a continuous inverse on X * . Moreover, the monotonicity of International Journal of Differential Equations Φ ′ on X * ensures that Φ is sequentially lower semicontinuously on X (see [33], Proposition 25. 20). e proof of the lemma is complete.
□ Lemma 3 (see [8]). Under assumptions (H1) and (H2), the functional Ψ is well defined and is of class C 1 on X. Moreover, its derivative is given by Moreover, Ψ 1 is compact from X to X * .

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.