A non-smooth Brezis-Oswald uniqueness result

Abstract We classify the non-negative critical points in W 0 1 , p ( Ω ) {W}_{0}^{1,p}\left(\Omega ) of xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> J v = ∫ H D − F x mathvariant="normal">d J\left(v)=\mathop{\int }\limits_{\Omega }\hspace{0.15em}H\left(Dv)-F\left(x,v){\rm{d}}x, where H is convex and positively p -homogeneous, while t width="0.33em" ↦ ∂ width="0.1em" / t\hspace{0.33em}\mapsto \hspace{0.33em}{\partial }_{t}F\left(x,t)\hspace{0.1em}\text{/}\hspace{0.1em}{t}^{p-1} non-increasing. Since may not be differentiable F has a one-sided growth condition, J only lower semi-continuous on . use weak notion point for non-smooth functionals, derive sufficient regularity latter without an Euler-Lagrange equation available, focus uniqueness part results study Brezis Oswald, using Picone inequality.