Preprint

A Liouville theorem for p-harmonic functions on exterior domains

E. N. Dancer, Daniel Daners, Daniel Hauer


Abstract

We prove Liouville type theorems for p-harmonic functions on an exterior domain Rd, where 1<p< and d2. If 1<p<d we show that every positive p-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions is constant. For pd and p1 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 any semi-bounded p-harmonic function is constant if 1<p<d. If pd then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous p-Laplace equation.

Keywords: elliptic boundary-value problems, Liouville-type theorems, p-Laplace operator, p-harmonic functions, exterior domain.

AMS Subject Classification: Primary 35B53,35J92,35B40.

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Friday, February 14, 2014