Positive solutions for elliptic problems with critical nonlinearity and combined singularity

Abstract

Consider a class of elliptic equation of the form $$ -\Delta u - {\lambda \over {|x|^2}}u = u^{2^\ast -1} + \mu u^{-q}\quad \mbox {in} \ \Omega \backslash \{0\} $$ with homogeneous Dirichlet boundary conditions, where $0\in \Omega \subset \mathbb R^N$($N\geq 3$), $0 < q < 1$, $0 < \lambda <(N-2)^2/4$ and $2^\ast = 2N/(N-2)$. We use variational methods to prove that for suitable $\mu $, the problem has at least two positive weak solutions.