Eigenvalues of the p(x)-biharmonic operator with indefinite weight

Abstract

In this article, we consider the nonlinear eigenvalue problem:

$$\left\{\begin{array}{ll}\Delta(|\Delta u|^{p(x)-2} \Delta u)=\lambda V(x)|u|^{q(x)-2}u,\quad{\rm in} \,\,\Omega\\ u=\Delta u=0, \qquad\qquad\qquad\quad\,\quad\,{\rm on}\,\,\partial \Omega,\end{array} \right.$$

where ${\Omega}$ is a bounded domain of ${\mathbb{R}^N}$ with smooth boundary, ${\lambda}$ is a positive real number, ${p,\, q: \overline{\Omega} \rightarrow (1,+\infty)}$ are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekeland’s variational principle.