Multiple Solutions for a Nonlocal Elliptic Problem Involving 
                     
                        
                           p
                           
                              
                                 
                                    x
                                 
                              
                              ,
                              q
                              
                                 
                                    x
                                 
                              
                           
                        
                     
                  -Biharmonic Operator

<jats:p>In this paper, using the variational principle, the existence and multiplicity of solutions for <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>p</mi>
                              <mrow>
                                 <mfenced open="(" close=")" separators="|">
                                    <mrow>
                                       <mi>x</mi>
                                    </mrow>
                                 </mfenced>
                                 <mo>,</mo>
                                 <mi>q</mi>
                                 <mfenced open="(" close=")" separators="|">
                                    <mrow>
                                       <mi>x</mi>
                                    </mrow>
                                 </mfenced>
                              </mrow>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>-Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.</jats:p>

As we all know, the harmonic equation and biharmonic equation in partial differential equation are the most widely used equations in the field of theoretical research and engineering application. Among them, the focus and difficulty in science and engineering is to solve the boundary value problem of biharmonic equation. erefore, many scholars have carried out a lot of in-depth research on biharmonic problems, see [1][2][3][4][5][6]. In particular, in [3], when the nonlinear functions F and G satisfy certain conditions, Li and Tang studied the (p, q)-biharmonic problem, by applying the three critical point theorem by Ricceri [7]. Two years later, Masser et al. [8] supposed that the assumption in problem (2) μ was 0 and imposed appropriate conditions on F. Based on the critical point theorem introduced by Bonanno and Molica Bisci, infinitely many solutions were obtained. It well known that the variable exponent case possess more complicated properties than the constant exponent case, and some methods used in the (p, q)-biharmonic case cannot be applied to the (p(x), q(x))-biharmonic case. erefore, Allaoui et al. [9] have made a great contribution to such problems, and they continued to extend (p, q)-biharmonic operator in [8] to (p(x), q(x))-biharmonic case, on the basis of Ricceri's variational principle [10] and the basic theory of Sobolev space, and the following system is solved: With the deepening of the investigation, scholars are increasingly paying attention to the solution of nonlocal elliptic equation, see [11][12][13][14][15], for details.
In [16], Ferrara et al. researched the problem: If the nonlinear term satisfies the certain conditions, then we obtain that there are at least two nontrivial solutions to problem (4) by using the variational method and mountain pass lemma. In the same year, Hssini et al. [17] studied the same problem as in [16] according to Ricceri's variational principle and obtained multiple solutions of problem (4). In addition, in [18], Xiao and Miao extended problem (4) and continued to study the multiplicity of solutions of (p 1 , p 2 , . . . , p n )-Kirchhoff type problems. In [19], the author extended the conclusion of the problem (4) to the p(x)-biharmonic operator.
So far, there are few results on the existence and multiplicity of solutions for (p(x), q(x))-Kirchhoff type problems under Navier boundary condition. erefore, inspired by the above research, in the present paper, our target is to show the existence and multiplicity of solutions of problem (1), according to the variational principle proposed by Ricceri and the basic results of Lebesgue or Sobolev spaces with a variable exponent.
is article consists of three sections. In Section 2, some basic conclusions and certain essential theorems or propositions of generalized Lebesuge-Sobolev spaces are obtained, which provides the significant framework for the research of variational problems and nonlocal elliptic problems with (p(x), q(x))-biharmonic operator. In Section 3, we have the main results of this article and gain the corresponding proof mainly according to the variational principle by Ricceri. Besides, at the end of Section 3, we give an example involving (p(x), q(x))-Kirchhoff type equations with Ω � [(−1, 1)] 2 , which is actually an application of eorem 2, that is to say, this example explains our main results very well.

Preliminaries
Define the space as follows: which has the norm e Sobolev space with variable exponents is defined as which has the norm (Ω). Besides, from [20], we know that L p(x) (Ω) and W k,p(x) (Ω) are Banach space and satisfy separability, uniform convexity, and reflexivity.
Define the functional I λ : X ⟶ R: Journal of Mathematics 3 Hence, according to Definition 1, we know that if (u, v) ∈ X is the weak solution of problem (1), it is equivalent to that (u, v) is the critical point of I λ . In addition, since the space X to C(Ω) × C(Ω) is a compact embedding, so Φ, Ψ: X ⟶ R are sequentially weakly lower semicontinuous, and it is clear that Φ is coercive.

Main Results and Proof
one has where 4 Journal of Mathematics en, for every Problem (1) has a sequence of solutions (u n , u n ) such that lim n⟶+∞ Φ(u n , u n ) � +∞.
According to eorem 1 (b), we know that eorem 2 is established.

Theorem 3. Presume (A 1 ) holds and
and one has where 8

Journal of Mathematics
en, for every problem (1) has infinity solutions converging to 0.
In the next step, we prove that I λ does not have the local minimum at 0. For λ ∈ Λ, we have Journal of Mathematics 9 and there exists a column of positive real numbers η n ⟶ 0 + as n ⟶ +∞ and θ > 0 such that Let u n (x) be defined by (38); then, we can obtain Combining (41), (51), and (52), we have Obviously, according to eorem 1 (c), we have completed the proof of Define the function F: where a 1 � 3 and a n+1 � n(a n ) 3 + 3(n ≥ 1) and R((a n+1 , a n+1 ), 1) is an open unit ball with its center point at (a n+1 , a n+1 ). It is clear that F ≥ 0 and F ∈ C 1 (R 2 ), for all n ∈ N + . e maximum of F on R((a n+1 , a n+1 ), 1) is F(a n+1 , a n+1 ) � (a n+1 ) 6 where |R(x 0 , O 1 )| is the measure of R(x 0 , O 1 ). At the time, ∀n ∈ N + , and we deduce ∀λ > 0; from eorem 2, we can infer that the equations, have a sequence of unbounded weak solutions.

Data Availability
No data were used to support this study.

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

Authors' Contributions
Each part of this paper is the result of the joint efforts of QZ and QM. ey contributed equally to the final version of the paper. All the authors have read and approved the final manuscript.