Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations

Abstract

We consider the Dirichlet problem for positive solutions of the equation −Δm (u) = f(u) in a bounded smooth domain Ω, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator, a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results, together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z≡{x∈ Ω ∨ D(u)(x) = 0 = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u ∈ C2(Ω∖{0}).