A p-Laplacian supercritical Neumann problem

TL;DR

This paper establishes the existence of positive, radially nondecreasing solutions to a supercritical p-Laplacian Neumann problem in the unit ball, analyzing their behavior as the nonlinearity exponent grows large.

Contribution

It introduces a variational approach to find solutions under mild assumptions on the nonlinearity, even in the supercritical regime, and studies their asymptotic behavior.

Findings

Existence of positive, radially nondecreasing solutions for supercritical nonlinearities.

Solutions exhibit specific asymptotic behavior as the exponent q approaches infinity.

Method applicable under mild conditions on the nonlinearity g.

Abstract

For $p>2$, we consider the quasilinear equation $-\Delta_p u+|u|^{p-2}u=g(u)$ in the unit ball $B$ of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case $g(u)=|u|^{q-2}u$, we detect the asymptotic behavior of these solutions as $q\to\infty$.