Radial solutions for the bilaplacian equation with vanishing or singular radial potentials

TL;DR

This paper investigates radial solutions to a bilaplacian equation with variable potentials, establishing existence results through compact embeddings in weighted Sobolev spaces.

Contribution

It introduces new existence results for radial solutions of the bilaplacian equation with variable potentials using compact embedding techniques.

Findings

Existence of radial solutions under certain conditions.

Use of compact embeddings of Sobolev spaces into weighted Lebesgue spaces.

Framework applicable to equations with vanishing or singular potentials.

Abstract

Given three measurable functions $V\left(r \right)\geq 0$, $K\left(r\right)> 0$ and $Q\left(r \right)\geq 0$, $r>0$, we consider the bilaplacian equation \[ \Delta^2 u+V(|x|)u=K(|x|)f(u)+Q(|x|) \quad \text{in }\,\mathbb{R}^N \] and we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces.