Maximum and anti-maximum principles for the p-Laplacian with a nonlinear boundary condition

Abstract

In this paper we study the maximum and the anti-maximum principles for the problem $Delta _{p}u=|u|^{p-2}u$ in the bounded smooth domain $Omega subset mathbb{R}^{N}$, with $|abla u|^{p-2}frac{partial u}{partial u }=lambda |u|^{p-2}u+h$ as a non linear boundary condition on $partial Omega $ which is supposed $C^{2eta }$ for some $eta $ in $]0,1[$, and where $hin L^{infty }(partial Omega )$. We will also examine the existence and the non existence of the solutions and their signs.

Keywords

• Anti-maximum
• p-laplacian
• non linear boundary condition.