Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

Abstract

We consider the existence of multiple solutions of the singular elliptic problem $-\mathrm{div}({|x|}^{-ap}{|\nabla u|}^{p-2}\nabla u)+{|u|}^{p-2}u/{|x|}^{(a+1)p}=f{|u|}^{r-2}u+h{|u|}^{s-2}u+{|x|}^{-b{p}^{*}}{|u|}^{p^{*}-2}u$, $u(x)\to 0$ as $|x|\to +\infty$, where $x\in {ℝ}^N$, , , $a\le b\le a+1,r$, $s>1$, $p^{*}=Np/(N-pd)$, $d=a+1-b$. By the variational method and the theory of genus, we prove that the above-mentioned problem has infinitely many solutions when some conditions are satisfied.