Symmetry of positive solutions of a p-Laplace equation with convex nonlinearites

Abstract

In this paper, we consider the symmetry properties of the positive solutions of a $ p $-Laplacian problem of the form $ \begin{eqnarray*} \begin{cases} -{{\Delta}}_p u = f(x,u),\ \ \ \ \ \ \ \mathrm{in}\ \ \ \ \ {{\Omega}},\\ \ \ \ \ \ \ \ u = g(x), \ \ \ \ \ \ \ \ \ \ \mathrm{on}\ \ \ \ \partial{{\Omega}}, \end{cases} \end{eqnarray*} $ where $ {{\Omega}} $ is an open smooth bounded domain in $ R^N, N\geq2 $, and symmetric w.r.t. the hyperplane $ T_0^\nu (\nu $ is a direction vector in $ R^N, |\nu| = 1 $$) $, $ f $: $ {{\Omega}}\times R^+\rightarrow R^+ $ is a continuous function of class $ C^1 $ w.r.t. the second variable, $ g\geq 0 $ is continuous, and both $ f $ and $ g $ are symmetric w.r.t. $ T^\nu_0 $, respectively. Introducing some assumptions on nonlinearities, we get that the positive solutions of the problem above are symmetric w.r.t. the direction $ \nu $ by a new simple idea even if $ {{\Omega}} $ is not convex in the direction $ \nu $.

Keywords

• symmetry of positive solutions
• $ p $-laplace equation
• convex nonlinearities
• weak maximum principle
• degenerate elliptic operator