Eigenvalues for a fourth order elliptic problem

Abstract:

We study the fourth order nonlinear eigenvalue problem with a $p(x)$-biharmonic operator \begin{equation*} \left \{\begin {array}{l} \Delta ^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda w(x)f(u)\quad \text {in}\ \Omega , \\ u=\Delta u=0\quad \text {on}\ \partial \Omega , \end{array} \right . \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$, $p\in C(\overline {\Omega })$ with $p(x)>1$ on $\overline {\Omega }$, $\Delta ^2_{p(x)}u=\Delta (|\Delta u|^{p(x)-2}\Delta u)$ is the $p(x)$-biharmonic operator, and $\lambda >0$ is a parameter. Under some appropriate conditions on the functions $p, a, w, f$, we prove that there exists $\overline {\lambda }>0$ such that any $\lambda \in (0,\overline {\lambda })$ is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle and some recent theory on the generalized Lebesgue–Sobolev spaces $L^{p(x)}(\Omega )$ and $W^{k,p(x)}(\Omega )$.