  We classify the non-negative critical points in $W^{1,p}_0(\Omega)$ of \[
J(v)=\int_\Omega H(Dv)-F(x, v)\, dx \] where $H$ is convex and positively
$p$-homogeneous, while $t\mapsto \partial_tF(x, t)/t^{p-1}$ is non-increasing.
Since $H$ may not be differentiable and $F$ has a one-sided growth condition,
$J$ is only l.s.c. on $W^{1,p}_0(\Omega)$. 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.
