Antimaximum principle in exterior domains. (January 2016) - Anoop, T.V.; Drábek, P.; Sankar, Lakshmi; Sasi, Sarath
Abstract: We consider the antimaximum principle for the p -Laplacian in the exterior domain: { − Δ p u = λ K ( x ) ∣ u ∣ p − 2 u + h ( x ) in  B 1 c, u = 0 on  ∂ B 1, where Δ p is the p -Laplace operator with p > 1, λ is the spectral parameter and B 1 c is the exterior of the closed unit ball in R N with N ≥ 1 . The function h is assumed to be nonnegative and nonzero, however the weight function K is allowed to change its sign. For K in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for h with compact support. For a compactly supported K, with N = 1 and p = 2, we provide a necessary and sufficient condition on h 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.