# American Institute of Mathematical Sciences

June  2017, 37(6): 3025-3057. doi: 10.3934/dcds.2017130

## A p-Laplacian supercritical Neumann problem

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

* Corresponding author

Received  November 2016 Revised  January 2017 Published  February 2017

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 $q\to \infty$.

Citation: Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
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##### References:
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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