The effect of domain topology on the number of positive solutions for singular elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities (Q2371846)

The goal of this paper is to consider the existence and multiplicity of positive solutions for the following nonlinear equation: \[ \begin{gathered} -\text{div}(|x|^{-2a}\nabla_x u)- \mu{u\over|x|^{2(1+ a)}}= \lambda{|u|^{q-2} u\over|x|^{dD}}+ {|u|^{p-2} u\over|x|^{bp}},\;x\in\Omega,\\ u> 0\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^d\) and contains \(0\) in its interior, \(0\leq\mu< (\sqrt{\mu_*}- a)^2\), \(\mu_*= ({d-2\over 2})^2\), \(d\geq 3\). Under some additional assumptions on the data of (1), the author using Lusternik-Schnirelmann category theory prove that (1) has at least \(\text{cat}(\Omega)\) positive solutions.