A p-Laplacian supercritical Neumann problem

Francesca Colasuonno; Benedetta Noris

doi:10.3934/dcds.2017130

Article Contents

2017, Volume 37, Issue 6: 3025-3057. Doi: 10.3934/dcds.2017130

This issue Previous Article Scattering of solutions to the nonlinear Schrödinger equations with regular potentials Next Article Onofri inequalities and rigidity results

A p-Laplacian supercritical Neumann problem

Francesca Colasuonno^, and
Benedetta Noris

Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine -CP214, boulevard du Triomphe, 1050 Bruxelles, Belgique

* Corresponding author
* Corresponding author

Received: November 2016

Revised: January 2017

Published: May 2017

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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 $q\to \infty$.

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Mathematics Subject Classification: Primary: 35J92, 35A15, 35B45; Secondary: 35B09, 35B40.

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Figure 1. The graph of the function $y=\left[\frac{p}{p-1}\left(\frac{x^p}p-\frac{x^q}q\right)\right]^{1/p}$ for $x,\,y\ge0$.

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