A Liouville theorem for \(p\)-harmonic functions on exterior domains

EN Dancer, Daniel Daners and Daniel Hauer
preprint (PDF), 28 January 2014
Positivity 19 (2015), 577–586
Original version appears at DOI 10.1007/s11117-014-0316-2
Citations on Google Scholar


We prove Liouville type theorems for \(p\)-harmonic functions on an exterior domain \(\mathbb R^{d}\), where \(1< p<\infty \) and \(d\geq 2\). If \(1< p< d \) we show that every positive \(p\)-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions is constant. For \(p\geq d\) and \(p\neq 1\) we show that positive \(p\)-harmonic functions are either constant or behave asymptotically like the fundamental solution of the \(p\)-Laplace operator. In the case of zero Neumann boundary conditions, we establish that every semi-bounded \(p\)-harmonic function is constant.

AMS Subject Classification (2000): 35B53,35J92,35B40

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