  We are interested in the following Dirichlet problem $$ \left\{
\begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu
\frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad
\mbox{in } \Omega \\ u = 0 & \quad \mbox{on } \partial \Omega, \end{array}
\right. $$ on a bounded domain $\Omega \subset \mathbb{R}^N$ with $0 \in
\Omega$. We assume that the nonlinear part is superlinear on some closed subset
$K \subset \Omega$ and asymptotically linear on $\Omega \setminus K$. We find a
solution with the energy bounded by a certain min-max level, and infinitely
many solutions provided that $f$ is odd in $u$. Moreover we study also the
multiplicity of solutions to the associated normalized problem.
