Antimaximum principle in exterior domains

We consider the antimaximum principle for the pp-Laplacian in the exterior domain: {−Δpu=λK(x)∣u∣p−2u+h(x)in  B1c,u=0on  ∂B1, where ΔpΔp is the pp-Laplace operator with p>1p>1,λλ is the spectral parameter and B1c is the exterior of the closed unit ball in RNRN with N≥1N≥1. The function hh is assumed to be nonnegative and nonzero, however the weight function KK is allowed to change its sign. For KK in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for hh with compact support. For a compactly supported KK, with N=1N=1 and p=2p=2, we provide a necessary and sufficient condition on hh for the global antimaximum principle. In the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems.