A Liouville theorem for p-harmonic functions on exterior domains

الأصل · EN

We prove Liouville type theorems for p-harmonic functions on exterior domains of the d-dimensional Euclidean space, where 1<p<∞ and d≥ 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≥ d, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous p-Laplace equation.