Radial nonlinear elliptic problems with singular or vanishing potentials

TL;DR

This paper establishes the existence of radial solutions for a class of nonlinear elliptic equations with singular or vanishing potentials by developing compact embeddings and applying variational methods.

Contribution

It introduces new compact embedding results for radial weighted Sobolev spaces and applies them to prove existence of solutions for nonlinear elliptic problems with singular or vanishing potentials.

Findings

Existence of radial solutions under certain conditions on potentials.

Development of compact embeddings for weighted Sobolev spaces.

Application of variational methods to nonlinear elliptic equations.

Abstract

In this paper we prove existence of radial solutions for the nonlinear elliptic problem \[ -\mathrm{div}(A(|x|)\nabla u)+V(|x|)u=K(|x|)f(u) \quad \text{in }\mathbb{R}^{N}, \] \noindent with suitable hypotheses on the radial potentials $A,V,K$. We first get compact embeddings of radial weighted Sobolev spaces into sum of weighted Lebesgue spaces, and then we apply standard variational techniques to get existence results.