A non-existence result for a nonlinear Neumann problem

TL;DR

This paper proves that under certain conditions, no solutions exist for a semilinear elliptic equation with nonlinear boundary conditions in large balls, using asymptotic analysis.

Contribution

It provides a new non-existence result for solutions of a nonlinear Neumann problem in large domains based on asymptotic analysis.

Findings

Non-existence of solutions for large radius R

Sufficient conditions for non-existence

Application of asymptotic analysis to boundary value problems

Abstract

In this note we consider a semilinear elliptic equation in $B_R$ with the nonlinear boundary condition, where $B_R$ is a ball of radius $R$. Under certain conditions, we establish a sufficient condition on the non-existence of solutions provided that $R$ is sufficiently large. The main argument is based on applying the asymptotic analysis to the equation with respect to $R\gg1$.