Quasilinear elliptic equations with singular potentials and bounded discontinuous nonlinearities

Abstract

In this paper we study the quasilinear equation \begin{equation} \begin{cases} - {\rm div}(|\nabla u|^{p-2} \nabla u)+V(|x|)|u|^{p-2} u= Q(|x|)f(u), & x\in \mathbb{R}^N, \\ u(x)\rightarrow 0,\quad |x|\rightarrow \infty. \end{cases} \tag{$\rm P$} \end{equation} with singular radial potentials $V,Q$ and bounded measurable function $f$. The approaches used here are based on a compact embedding from the space $W^{1,p}_r(\mathbb{R}^N; V)$ into $L^1 (\mathbb{R}^N; Q)$ and a new multiple critical point theorem for locally Lipschitz continuous functionals.