Existence of solutions to a slightly supercritical pure Neumann problem

TL;DR

This paper proves the existence and multiplicity of sign-changing solutions to a slightly supercritical Neumann boundary value problem in a ball, using symmetry and Lyapunov-Schmidt reduction techniques.

Contribution

It is the first to establish existence results for supercritical Neumann problems, extending to symmetric domains and incorporating zero-average conditions.

Findings

Existence of solutions in supercritical regime

Solutions must change sign due to zero-average condition

Method applicable to symmetric domains like ellipsoids

Abstract

We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problems in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric domains (e.g. ellipsoids) are also discussed.