Eigenvalues of some $p(x)$-biharmonic problems under Neumann boundary conditions

Abstract

In this paper, we study the following $p(x)$-biharmonic problem in Sobolev spaces with variable exponents \begin{equation} \begin{cases} \triangle ^{2}_{p(x)}u=\lambda ({\partial F } (x,u)/{\partial u}) & x\in \Omega , \\ {\partial u}/{\partial n}=0 & x\in \partial \Omega ,\\ {\partial }(|\triangle u|^{p(x)-2}\triangle u)/{\partial n} =a(x)|u|^{p(x)-2}u & x\in \partial \Omega . \end{cases} \end{equation} By means of the variational approach and Ekeland's principle, we establish that the above problem admits a nontrivial weak solution under appropriate conditions.