The Multiplicity of Nontrivial Solutions for a New 
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                  -Kirchhoff-Type Elliptic Problem

<jats:p>In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a <jats:inline-formula>
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                  </jats:inline-formula>-Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new <jats:inline-formula>
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                  </jats:inline-formula>-Kirchhoff problem has at least two nontrivial weak solutions.</jats:p>

The study of variational problems with nonstandard pðxÞ -growth conditions has been a new and interesting topic. These problems arise from the image processing model and stationary thermorheological viscous flows; we can refer to [1,2]. Problem (1) is related to the stationary problem where ρ, h, δ, p 0 , L are constants which represent some physical meanings; E is Young's modulus; uðx, tÞ is the lateral displacement; and f 1 , f 2 are the external forces. When f 1 = f 2 , it extends the classical D'Alembert wave equation for free vibrations of elastic strings. Since problem (2) is no longer a pointwise identity, it is often called a nonlocal problem. In recent years, nonlocal elliptic problems have attracted wide attention, and some important and interesting results have been established (see [3][4][5]).
In the past few decades, many people studied the following pðxÞ-Kirchhoff problem: where Ω be a bounded open subset of ℝ N with a C 1 -boundary ∂Ω, MðtÞ ∈ C 1 ð½0,+∞ÞÞ is a bounded below function, f : Ω × ℝ ⟶ ℝ is a Carathéodory function. For example, Dai in [4] using the three-critical-point theorem obtained the existence of solutions for problem (3). Moreover, when f ðx, uÞ = λðaðxÞjuj qðxÞ−2 u + bðxÞjuj rðxÞ−2 uÞ, Zhou and Ge in [6] studied the existence of the solution for the nonlocal problem (3) by using the fibering map approach for the corresponding Nehari manifold. In recent years, some people are starting to pay attention to the case MðtÞ = a − b tða ≥ b > 0Þ in problem (3). Obviously, MðtÞ = a − btða ≥ b > 0Þ is not bounded below. Therefore, this is a new class of nonlocal problems. In fact, for the case pðxÞ ≡ p, some results are given in [7][8][9]. However, few literatures have considered this new nonlocal pðxÞ-Kirchhoff problem. Recently, the authors in [10] have considered the following pðxÞ-Kirchhofftype problem: where a ≥ b > 0 are constants, Ω ∈ ℝ N is a bounded smooth domain, p ∈ Cð ΩÞ with 1 < pðxÞ < N, λ is a real parameter, and g is a continuous function. Under appropriate hypotheses, the author used a mountain pass theorem and fountain theorem to obtain the existence and multiplicity of nontrivial solutions for problem (4).
Inspired by the above facts and aforementioned papers, the main purpose of this paper is to study the existence of two nontrivial solutions for problem (1). Before stating our main results, we make some assumptions on the functions c, d, p, and r.

Preliminaries
In order to discuss problem (1), we need some theories on spaces L pðxÞ ðΩÞ and W 1,pðxÞ ðΩÞ which we will call generalized Lebesgue-Sobolev spaces. For more details on the basic properties of these spaces, we refer the readers to Fan and Zhao [11]. Set C + ð ΩÞ = fpðxÞ ; pðxÞ ∈ Cð ΩÞ, pðxÞ > 1,∀x ∈ Ωg; ζð ΩÞ denoted the set of all measurable real functions defined on Ω.
For any p ∈ C + ð ΩÞ, the variable exponent Lebesgue space L pðxÞ ðΩÞ is defined as with the norm The variable exponent Sobolev space W 1,pðxÞ ðΩÞ is defined as with the norm Define W 1,pðxÞ 0 ðΩÞ as the closure of C ∞ 0 ðΩÞ in W 1,pðxÞ ðΩÞ. From [11], we know that the spaces L pðxÞ ðΩÞ and W 1,pðxÞ ðΩÞ are all separable and reflexive Banach spaces.
Moreover, there is a constant C > 0, such that for all u ∈ W ðΩÞ. In addition, we can recall the following properties of the variable exponent spaces.

Proof of Main Result
In this section, the existence of nontrivial solutions for problem (1) is obtained by using a mountain pass lemma combined with Ekeland's variational principle.

Lemma 9.
Assume that the conditions of Theorem 1 hold. Then, the function J λ satisfies the ðPSÞ c condition with c < a 2 /2b for λ > 0 small enough.
Proof. Firstly, we prove that fu n g is bounded in W 1,pðxÞ 0 ðΩÞ.
Let fu n g ⊂ W 1,pðxÞ 0 ðΩÞ be a ðPSÞ c sequence such that c < a 2 / 2b. Arguing by contradiction, we assume that ku n k ⟶ +∞ as n ⟶ +∞.
So, one has c 2 < a 2 /2b. Letting fu n g be a ðPSÞ c 2 sequence of J λ , by Lemma 9, we obtain J λ which satisfies the ðPSÞ c 2 condition.

Data Availability
The findings in this research do not make use of data.

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