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

A Liouville theorem for $$p$$ p -harmonic functions on exterior domains We prove Liouville type theorems for $$p$$ p -harmonic functions on exterior domains of $${\mathbb {R}}^{d}$$ R d , where $$1<p<\infty$$ 1 < p < ∞ and $$d\ge 2$$ d ≥ 2 . We show that every positive $$p$$ p -harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $$|x|$$ | x | tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $$p$$ p -harmonic function is constant if $$1<p<d$$ 1 < p < d . If $$p\ge d$$ p ≥ d , then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $$p$$ p -Laplace equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 19 (3) – Dec 10, 2014
10 pages

/lp/springer_journal/a-liouville-theorem-for-p-p-harmonic-functions-on-exterior-domains-0NQHUg2hI2
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0316-2
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

PositivitySpringer Journals

Published: Dec 10, 2014

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