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

20 pages. To appear on a special issue of Rendiconti Lincei: Matematica ed Applicazioni dedicated to Antonio Ambrosetti