Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence

TL;DR

This paper establishes existence and multiplicity of finite energy solutions for nonlinear elliptic equations in radial Sobolev spaces, accommodating various potentials and nonlinearities, including superlinear and sublinear cases.

Contribution

It provides new existence and multiplicity results for elliptic equations with general potentials and nonlinearities in radial domains, extending previous work to more general conditions.

Findings

Existence of solutions in bounded and unbounded radial domains.

Multiplicity results for solutions under various nonlinearities.

Handling of potentials that vanish or are unbounded at zero or infinity.

Abstract

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }\Omega \subseteq \mathbb{R}^{N},\ N\geq 3, \] where $\Omega $ is a radial domain (bounded or unbounded) and $u$ satisfies $u=0$ on $\partial \Omega $ if $\Omega \neq \mathbb{R}^{N}$ and $u\rightarrow 0$ as $\left| x\right| \rightarrow \infty $ if $\Omega $ is unbounded. The potential $V$ may be vanishing or unbounded at zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear. If $g$ is sublinear, the case with $g\left( \left| \cdot \right| ,0\right) \neq 0$ is also considered.