A Liouville theorem for $$p$$ p -harmonic functions on exterior domains

Abstract

ABSTRACT. We prove Liouville type theorems for p-harmonic functions on exterior domains of R d , 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