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

Abstract:

We introduce and study an approximate solution of the $p$-Laplace equation and a linearlization $\mathcal {L}_{\epsilon }$ of a perturbed $p$-Laplace operator. By deriving an $\mathcal {L}_{\epsilon }$-type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly $p$-harmonic functions with finite $p$-energy on a complete noncompact manifold $M$ 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 $M$ has at most one $p$-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly $p$-harmonic functions with finite $q$-energy on Riemannian manifolds. As an application, we extend this theorem to some $p$-harmonic maps such as $p$-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for $p$-harmonic morphisms.