A non-smooth Brezis-Oswald uniqueness result

We classify the non-negative critical points in W-0(1,p)(Omega) ofJ(nu) =integral H-Omega(D nu) - F(x, nu)dx,where H is convex and positively p-homogeneous, while t bar right arrow partial derivative F-t(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 lower semi-continuous on W-0(1,p)(Omega). We use 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 the study of Brezis and Oswald, using a non-smooth Picone inequality.