Multiple solutions for a problem with resonance involving the $p$-Laplacian

Abstract

In this paper we will investigate the existence of multiple solutions for the problem $$\left(P\right)-{\Delta }_{p}u+g\left(x,u\right)={\lambda }_{1}h\left(x\right){\left|u\right|}^{p-2}u,\text{in}\Omega ,u\in H_0^{1,p}\left(\Omega \right)$$ where ${\Delta }_{p}u=\text{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)$ is the $p$-Laplacian operator, $\Omega \subseteq {ℝ}^N$ is a bounded domain with smooth boundary, $h$ and $g$ are bounded functions, $N\ge 1$ and . Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).