Existence and Multiplicity of Solutions for Choquard Type Problem Involving p(x)-Biharmonic Operator

In this paper, we are interested in the existence and multiplicity of solutions for the following Choquard type problem involving p(x)-biharmonic operator \(\begin{aligned} \left\{ \begin{array}{ll} M \left( \int _{\Omega }\frac{1}{p(x)}|\Delta u|^{p(x)}dx \right) \Delta _{p(x)}^{2}u-\Delta _{p(x)}u& \\ \qquad =\lambda \left( \int _{\Omega }\frac{|u(y)|^{q(y)}}{|x-y|^{\alpha (x,\,y)}}dy\right) |u|^{q(x)-2}u+|u|^{p^{*}(x)-2}u,\\ u=\Delta u=0 \ \ \ \text {on} \ \ \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}M\left({\int }_{\mathrm{\Omega }},\frac{1}{p(x)},{|\mathrm{\Delta }u|}^{p(x)},d,x\right){\mathrm{\Delta }}_{p(x)}^{2}u-{\mathrm{\Delta }}_{p(x)}u\hfill & \\ =\lambda \left({\int }_{\mathrm{\Omega }},\frac{{|u(y)|}^{q(y)}}{{|x-y|}^{\alpha (x,y)}},d,y\right){|u|}^{q(x)-2}u+|u|_{}^{p_{}^{\ast }(x)-2}u,\hfill \\ u=\mathrm{\Delta }u=0\text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded domain of \({\mathbb {R}}^{N}(N\ge 3)\) ${\mathbb{R}}^N(N\ge 3)$ with a Lipschitz boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ , \(\Delta _{p(x)}^{2}u:=\Delta (|\Delta u|^{p(x)-2}\Delta u)\) ${\mathrm{\Delta }}_{p(x)}^{2}u:={\mathrm{\Delta }(|\mathrm{\Delta }u|}^{p(x)-2}\mathrm{\Delta }u)$ is the p(x)-biharmonic operator, \(p(x)<q(x)<p^{*}(x):=\frac{Np(x)}{N-2p(x)}\) $p(x)<q(x)<p_{}^{\ast }(x):=\frac{Np(x)}{N-2p(x)}$ for all \(x\in {\overline{\Omega }}\) $x\in \overline{\mathrm{\Omega }}$ , \(M:\,{\mathbb {R}}_{0}^{+}:=[0,\,+\infty )\rightarrow (0,\,+\infty )\) $M:{\mathbb{R}}_0^{+}:=[0,+\infty )\to (0,+\infty )$ is a nondecreasing and continuous function and \(\lambda >0\) $\lambda >0$ is a parameter. We make use of the concentration-compactness principle and variational methods to obtain the multiplicity of solutions both in non-degenerate and degenerate case.