Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

Abstract

We introduce and study an approximate solution of the p p -Laplace equation and a linearlization L ϵ \mathcal {L}_{\epsilon } of a perturbed p p -Laplace operator. By deriving an L ϵ \mathcal {L}_{\epsilon } -type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly p p -harmonic functions with finite p p -energy on a complete noncompact manifold M M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M M has at most one p p -hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p p -harmonic functions with finite q q -energy on Riemannian manifolds. As an application, we extend this theorem to some p p -harmonic maps such as p p -harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p p -harmonic morphisms.