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

TL;DR

This paper investigates the existence and multiplicity of solutions for a nonlinear elliptic boundary value problem with singular potentials at the origin and boundary, using variational methods and symmetry assumptions.

Contribution

It introduces new existence and multiplicity results for elliptic problems with mixed nonlinearities and singular potentials, including solutions with bounded energy and infinitely many solutions under symmetry.

Findings

Existence of solutions with energy below a certain min-max level.

Infinitely many solutions when the nonlinearity is odd in u.

Analysis of solution multiplicity for the normalized problem.

Abstract

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.