PreprintA Liouville theorem for \(p\)harmonic functions on exterior domainsE. N. Dancer, Daniel Daners, Daniel HauerAbstractWe 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 any semibounded \(p\)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. Keywords: elliptic boundaryvalue problems, Liouvilletype theorems, \(p\)Laplace operator, \(p\)harmonic functions, exterior domain.AMS Subject Classification: Primary 35B53,35J92,35B40.
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