Uniqueness of the critical point for solutions to some $p$-Laplace equations in the plane

TL;DR

This paper proves that positive quasi-concave solutions to certain p-Laplace equations in convex planar domains have a unique critical point, leading to strict concavity of specific transformations of these solutions.

Contribution

It establishes the uniqueness of the critical point for solutions to a class of p-Laplace equations in convex planar domains, a novel result in this context.

Findings

Solutions have only one critical point.

Transformations of solutions are strictly concave.

Results apply to convex bounded domains in the plane.

Abstract

We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions.