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

Shu Cheng Chang, Jui Tang Chen, Shihshu Walter Wei

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒϵ of a perturbed p-Laplace operator. By deriving an ℒϵ-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.

Original languageEnglish
Pages (from-to)787-825
Number of pages39
JournalTransactions of the American Mathematical Society
Issue number2
Publication statusPublished - 2016 Feb


  • Liouville type properties
  • Perturbed p-Laplace operator
  • Weakly p-harmonic function
  • p-harmonic map
  • p-hyperbolic end

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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