Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on $\mathbb {R}^N$

Abstract:

We study weighted Sobolev type embeddings of radially symmetric functions from $W_r^{1,p}(\mathbb {R}^N; V)$ into $L^q(\mathbb {R}^N; Q)$ for $q<p$ with singular potentials. We then investigate the existence of nontrivial radial solutions of quasilinear elliptic equations with singular potentials and sub-$p$-linear nonlinearity. The model equation is of the form \[ \begin {cases} -\hbox {div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)|u|^{q-2} u, \quad x\in \mathbb {R}^N,\\ u(x)\rightarrow 0, \quad |x|\rightarrow \infty .\end {cases}\]