Nonlinear superharmonic functions and supersolutions

Abstract

Due to the lack of representation formulas for superharmonic functions associated with p-harmonic equations \({-\nabla \cdot(|\nabla u|^{p-2}\nabla u) = \mu}\) and their generalizations \({-\nabla \cdot A(x,\nabla u) = \mu}\),where \({A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^{p}}\), the interplay between nonlinear superharmonic functions and supersolutions is more important than in the linear case. Using the recent result of Kilpelainen et. al., we establish sufficient and necessary conditions in terms of the Riesz measure μ that a p-superharmonic function is an ordinary weak supersolution. As an example we consider p-superharmonic solutions of the Poisson-type equation \({-\nabla \cdot A(x,\nabla u) = f(x)}\).