Abstract
We prove Liouville type theorems for \(p\)-harmonic functions on exterior domains of \({\mathbb {R}}^{d}\), where \(1<p<\infty \) and \(d\ge 2\). 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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Acknowledgments
We thank the referees for the careful reading and pointing out some mistakes in an earlier version of the manuscript. Also we thank for the suggestion to state Lemma 3.2 explicitely.
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Dancer, E.N., Daners, D. & Hauer, D. A Liouville theorem for \(p\)-harmonic functions on exterior domains. Positivity 19, 577–586 (2015). https://doi.org/10.1007/s11117-014-0316-2
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DOI: https://doi.org/10.1007/s11117-014-0316-2
Keywords
- Elliptic boundary-value problems
- Liouville-type theorems
- \(p\)-Laplace operator
- \(p\)-Harmonic functions
- Exterior domain