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In this article, we study the following problem with Navier boundary conditions $$\displaylines{ \Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u =\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\cr u=\Delta u=0 \quad \text{on } \partial\Omega. }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega$, $N\geq1$. $p(x),q(x)$ and $\gamma(x)$ are continuous functions on $\overline{\Omega}$, $\lambda$ and $\mu$ are parameters. Using variational methods, we establish some existence and non-existence results of solutions for this problem.