Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain

We are interested in the following Dirichlet problem $$\{\begin{array}{ll}-\mathrm{\Delta }u+\lambda u-\mu \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^2}-\nu \frac{u}{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,{\mathbb{R}}^N\setminus \mathrm{\Omega }{)}^2}=f(x,u)& in\mathrm{\Omega }\\ u=0& on\mathrm{\partial }\mathrm{\Omega },\end{array}$$ on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with $0\in \mathrm{\Omega }$. We assume that the nonlinear part is superlinear on some closed subset $K\subset \mathrm{\Omega }$ and asymptotically linear on $\mathrm{\Omega }\setminus K$. 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.