Multiple positive solutions for a class of nonlinear elliptic equations (Q596728)

The present study deals with the existence and multiplicity of positive solutions of the following problem: \[ \begin{cases} -\Delta u-\frac {\gamma}{|x|^2}u=u^p +\lambda u^q\quad &\text{in }\Omega \setminus\{0\}\\ u(x)>0\quad &\text{in } \Omega\setminus\{0\}\\ u(x)=0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(0 \in\text{int}\,\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is a bounded domain with smooth boundary, \(0\leq\gamma<\overline\gamma=\left({N-2} {2} \right)^2\) and \(\overline\gamma\) is the best constant in the Hardy inequality. Moreover, \(p=2^*-1\), where \(2^*=\frac{2N}{N-2}\) is the so-called critical Sobolev exponent and \(0<q<1\).