We present some results concerning the problem $-\Delta u=\lambda \frac{u}{{\left|x\right|}^2}+u^q$, $u>0$ in $\Omega$, ${u|}_{\partial \Omega }=0$, where $0<q<(N+2)/\left(N-2\right)$, $q\ne 1$, $\lambda \ge 0$ and $\Omega$ is a smooth bounded domain containing the origin. In particular, bifurcation and uniqueness results are discussed.

We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected $C$ ($\supset$-shape). Next, we show that there is a relationship between these two problems.