On bifurcation and uniqueness results for some semilinear elliptic equations involving a singular potential

We will present some results concerning the problem \begin{equation}\label{eq:sl} \left\{ \begin{array}{ll} & -\D u = \lambda \dfrac {u}{|x|^2}+u^q \, , \, \, u > 0 \hbox{ in }\O,\\ & u|_{\p \O}=0, \end{array} \right. \end{equation} where $0