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arXiv · 1411.1492

Liouville type theorems for the p-harmonic functions

Jingyi Chen·Yue Wang
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Abstract

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, we find an incomplete Riemannian metric on ${\mathbb R}^2$ with positive Gauss curvature such that every positive p-harmonic function must be constant for $p\geq 4$.

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Jingyi Chen, Yue Wang. 2014-11-06. Liouville type theorems for the p-harmonic functions. https://doi.org/10.2140/pjm.2016.282.313

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