On quasilinear Brezis-Nirenberg type problems with weights

Abstract

In this paper we study Brezis-Nirenberg type results for radial solutions of a quasilinear elliptic equation of the form $$ \begin{cases}-\Delta_pu= \lambda C(|x|)|u|^{p-2}u+ B(|x|) |u|^{q-2}u, \ a.e.\ x\in B_R(0)\subset\mathbb R^N,\ R>0,\\ u=0,\quad \mbox{on }\partial B_R(0), \end{cases} $$ where $\lambda\in\mathbb R$, $q\ge p>1$, $\Delta_pu=\mbox{div}(|\nabla u|^{p-2}\nabla u)$, $B_R(0)$ denotes the ball of radius $R>0$ centered at $0$ in $\mathbb R^N$, and the weights $ B,\ C$ are appropriate positive measurable radially symmetric functions.