Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a \(p(x)\)-biharmonic operator \[\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\in C(\overline{\Omega})\), \(\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)\) is the \(p(x)\)-biharmonic operator, and \(\lambda\gt 0\) is a parameter. We establish sufficient conditions under which there exists a positive number \(\lambda^{*}\) such that the above problem has at least two nontrivial weak solutions for each \(\lambda\gt\lambda^{*}\). Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\).