Antimaximum principle in exterior domains. (January 2016)

• Record Type: Journal Article
• Authors: Anoop, T.V.; Drábek, P.; Sankar, Lakshmi; Sasi, Sarath
• Is Part Of: Nonlinear analysis. Volume 130(2015)
• Page Start: 241
• Page End: 254
• Publication Date: 2016-01
• Subjects: 35J92 -- 35P30 -- 35B05; Exterior domains -- p-Laplacian -- Local and global antimaximum principle -- Positive eigenfunctions -- Regularity results; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.10.010
• Languages: English
• ISSNs: 0362-546X

Abstract

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.