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

Abstract

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -Δu + λu - μ\frac{u}{|x|^2} - ν\frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus Ω)^2} = f(x,u) & \quad \mbox{in } Ω\\ u = 0 & \quad \mbox{on } \partial Ω, \end{array} \right. $$ 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.