A non-smooth Brezis-Oswald uniqueness result

Abstract

We classify the non-negative critical points in $W_0^{1,p}(\Omega )$ of \[ J(v)=\int_ΩH(Dv)-F(x, v)\, dx \] where $H$ is convex and positively $p$-homogeneous, while $t\mapsto {\partial }_{t}F(x,t)/t^{p-1}$ is non-increasing. Since $H$ may not be differentiable and $F$ has a one-sided growth condition, $J$ is only l.s.c. on $W_0^{1,p}(\Omega )$. We employ a weak notion of critical point for non-smooth functionals, derive sufficient regularity of the latter without an Euler-Lagrange equation available and focus on the uniqueness part of the results in \cite{BO}, through a non-smooth Picone inequality.

30 pages, comments welcome!