Multiple Solutions for p-Laplacian Type Problems with an Asymptotically p-linear Term

Abstract

The aim of this paper is studying the asymptotically p-linear problem $\left\{\begin{array}{clclcl}{\rm{-div}(A(x,u)|\bigtriangledown|_{u}|^{p-2}\bigtriangledown u)+ \frac{1}{p}A_{t}(x,u)|\bigtriangledown u|^{p}} \\ {\qquad = \; \lambda|u|^{p-2}u+g(x,u) \qquad \qquad \qquad \rm {in} \Omega} \\ {u=0 \qquad \qquad\qquad\qquad \qquad \qquad\qquad\qquad\rm{on}\; \partial \Omega},\end{array} \right.$ where $\Omega \subset \mathbb{R}^{N}$ is an open bounded domain and $p > N \geq 2$. Suitable assumptions both at infinity and in the origin on the even function A(x, ·) and the odd map g(x, ·) allow us to prove the existence of multiple solutions by means of variational tools and the pseudo-index theory related to the genus in $W^{1,p}_{0}(\Omega)$.

Mathematics Subject Classification (2010). 35J60, 35J35, 47J30, 58E05.