Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions

Abstract

In this work we will study the eigenvalues for a fourth order elliptic equation with p(x)-growth conditions ∆ 2 p(x) u = λ|u| p(x)−2 u, under Neumann boundary conditions, where p(x) is a continuous function defined on the bounded domain with p(x) > 1. Through the Ljusternik-Schnireleman theory on C 1 -manifold, we prove the existence of infinitely many eigenvalue sequences and sup Λ = +∞, where Λ is the set of all eigenvalues.