Multiple nonradial solutions for a nonlinear elliptic radial problem: an improved result

TL;DR

This paper improves existing results on the existence of multiple nonradial solutions to a nonlinear elliptic equation involving singular weights, expanding the class of solutions known for such problems.

Contribution

It provides an enhanced existence theorem for nonnegative nonradial solutions to a weighted nonlinear elliptic problem, extending previous results with weaker assumptions.

Findings

Established the existence of multiple nonradial solutions.

Extended the class of nonlinearities for which solutions exist.

Improved the conditions under which solutions are guaranteed.

Abstract

We obtain an improved version of a recent result concerning the existence of nonnegative nonradial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left| x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\displaystyle\frac{A}{\left| x\right| ^{\alpha }}u=f\left( u\right) \quad \text{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,\alpha >0, \] where $f$ is a continuous nonlinearity satisfying $f\left( 0\right) =0$.