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arXiv · 1411.4224

A Liouville theorem for $p$-harmonic functions on exterior domains

E. N. Dancer·Daniel Daners·Daniel Hauer
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Abstract

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$ tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $p$-harmonic function is constant if $1<p<d$. If $p\ge d$, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $p$-Laplace equation.

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E. N. Dancer, Daniel Daners, Daniel Hauer. 2014-11-16. A Liouville theorem for $p$-harmonic functions on exterior domains. https://doi.org/10.1007/s11117-014-0316-2

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