A Liouville theorem for -harmonic functions on exterior domains
E. N. Dancer, Daniel Daners, Daniel Hauer
Abstract
We prove Liouville type theorems for -harmonic functions on
an exterior domain , where
and
. If
we show that every positive
-harmonic function satisfying zero Dirichlet, Neumann or
Robin boundary conditions is constant. For and
we show that positive -harmonic functions are
either constant or behave asymptotically like the fundamental
solution of the -Laplace operator. In the case of zero
Neumann boundary conditions, we establish that any
semi-bounded -harmonic function is constant if
. If then it is either constant or it
behaves asymptotically like the fundamental solution of the
homogeneous -Laplace equation.
Keywords:
elliptic boundary-value problems, Liouville-type theorems, -Laplace operator, -harmonic functions, exterior domain.
AMS Subject Classification:
Primary 35B53,35J92,35B40.