Summary: 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 \(\lambda> 0\) is a parameter. We establish sufficient conditions under which there exists a positive number \(\lambda^\ast\) such that the above problem has at least two nontrivial weak solutions for each \(\lambda >\lambda^\ast\). 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 \(W^{k,p(x)}(\Omega)\).