A non-smooth Brezis-Oswald uniqueness result

Abstract

We classify the non-negative critical points in W01,p(Ω){W}_{0}^{1,p}\left(\Omega ) of J(v)=∫ΩH(Dv)−F(x,v)dx,J\left(v)=\mathop{\int }\limits_{\Omega }\hspace{0.15em}H\left(Dv)-F\left(x,v){\rm{d}}x, where HH is convex and positively pp-homogeneous, while t↦∂tF(x,t)/tp−1t\hspace{0.33em}\mapsto \hspace{0.33em}{\partial }_{t}F\left(x,t)\hspace{0.1em}\text{/}\hspace{0.1em}{t}^{p-1} is non-increasing. Since HH may not be differentiable and FF has a one-sided growth condition, JJ is only lower semi-continuous on W01,p(Ω){W}_{0}^{1,p}\left(\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.

Keywords

• non-smooth critical point theory
• regularity theory
• uniqueness of solutions to nonlinear pde
• differential inclusions
• picone inequality
• 35j62
• 35b65
• 35a02
• 35p30