A non-smooth Brezis-Oswald uniqueness result

We classify the non-negative critical points in $W_0^{1,p}(\mathrm{\Omega })$ of $$J(v)={\int }_{\mathrm{\Omega }}H(Dv)-F(x,v)dx$$ where $H$ is convex and positively $p$-homogeneous, while $t\mapsto {\mathrm{\partial }}_{t}F(x,t)\mathrm{/}{t}^{p-1}$ is non-increasing. Since $H$ may not be differentiable and $F$ has a one-sided growth condition, $J$ is only l.s.c. on $W_0^{1,p}(\mathrm{\Omega })$. 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.