Radial solutions for a quasilinear equation via Hardy inequalities

Abstract

We establish an analogue of the Sobolev critical exponent for the inclusion $V^p(a)\hookrightarrow L^q(b)$, where $a$ and $b$ are weight functions, $V^p(a)$ is a weighted Sobolev space, and $L^q(b)$ is a weighted Lebesgue space. We use this result to study existence of radial solutions to the problem with weights $$(D_w)\quad\quad\begin{cases} -{\rm div}(\tilde{a}(|x|)|\nabla u|^{p-2}\nabla u)= \tilde{b}(|x|)|u|^{q-2}u \qquad \mbox{in} \quad \Omega\subset \mathbb R^N ,\\~~u=0\qquad \mbox{on}\quad \partial\Omega, \end{cases} $$ where $\Omega$ is a ball, $1 <p <q,$ and $\tilde{a}(|x|)=|x|^{1-N}a(|x|),$ $\tilde{b}(|x|)=|x|^{1-N}b(|x|)$. We are interested in the interplay between $q$ and a suitable critical exponent and its consequences for the existence and nonexistence of positive solutions of problem $(D_w).$