Positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents (Q908223)

In this paper, the following semilinear elliptic problem \[ \begin{cases} -\operatorname{div}(|x|^{-2a}\nabla u)-\mu\frac{u}{|x|^{2(1+a)}}=\frac{|u|^{p-2}}{|x|^{bp}}u+f(x,u),\quad & x\in\Omega, \\ u= 0,\quad & x\in\partial\Omega\end{cases} \] is considered, where \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary \(\partial \Omega\), \(N\geq 3\), \(0\in\Omega\), \(0\leq a<\sqrt{\overline{\mu}}\), \(0\leq\mu<(\sqrt{\overline{\mu}}-a)^2\) with \(\overline{\mu}=(N-2)^2/4\), \(a\leq b<a+1\), and \(p=2N/(N-2(1+a-b))\) is the Hardy-Sobolev critical exponent. The existence result of positive solutions is established.