Regularity and asymptotic approach to semilinear elliptic equations with singular potential

We study weak solutions of the problem $$ \begin{dcases*} \ - \Delta u = \frac{\lambda}{|x|^2} u + u^p & \ \ \ in \ $\Omega \backslash\{0\}$\\ \ u \geq 0 & \ \ \ in \ $\Omega \backslash\{0\}$\\ \ u|_{\partial \Omega} =0 & \end{dcases*} $$ where $\Omega \subseteq \real^N$ is a smooth bounded domain containing the origin, $N \geq 3$, $1 < p < (N+2) / (N-2)$ and $-\infty < \lambda \leq \bar{\lambda} : = (N-2)^2/4$. We present a regularity estimate around the origin, generalizing previous results of other authors for the case $\lambda \geq 0$. For the case of radially symmetric solutions on the unit ball $\Omega = B_1(0)$ we present a very good approximation for the shape of the solution in the limit when $\lambda \to -\infty$.