Combined effects of singular and critical nonlinearities in elliptic problems (Q393206)

The paper deals with existence and multiplicity of positive weak solutions to a class of elliptic problems with the nonlinearity containing both singular and critical terms NEWLINE\[NEWLINE \begin{cases} \Delta u=\lambda u^{2^*-1}+p(x)u^{-\gamma} & \text{in } \Omega,\\ u=0 & \text{on }\partial\Omega, \end{cases} NEWLINE\]NEWLINE where \(\Omega\subset{\mathbb R}^N\) is a bounded domain, \(N\geq3,\) \(\lambda>0\) and \(\gamma\in(0,1)\) are constants, \(2^*\) is the critical Sobolev exponent and \(p: \Omega\to {\mathbb R}\) is a given non-negative function in \(L^2(\Omega)\). By means of the concentration-compactness principle due to Lions and the Ekeland variational principle, two positive weak solutions of the problem considered are obtained.