Quasilinear elliptic equations on $${\mathbb{R}^{N}}$$ with singular potentials and bounded nonlinearity

Abstract

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation $$\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.$$ with singular radial potentials V,Q and bounded nonlinearity f. The approaches used here are based on a compact embedding from \({W_r^{1,p}(\mathbb{R}^N; V)}\) into \({L^1(\mathbb{R}^N; Q)}\) and minimax methods. A uniqueness result is given for f ≡ 1.