A priori bounds for superlinear elliptic equations with semidefinite nonlinearity. (March 2017)

• Record Type: Journal Article
• Authors: Naito, Yūki; Suzuki, Takashi; Toyota, Yohei
• Is Part Of: Nonlinear analysis. Volume 151(2017)
• Page Start: 18
• Page End: 40
• Publication Date: 2017-03
• Subjects: primary 35B45 35B53 35J60 -- secondary 35K55; A priori estimates -- Liouville-type theorem -- Nonlinear elliptic equation -- Method of moving planes -- Nonlinear parabolic equation; 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.2016.11.016
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We derive a priori bounds for positive solutions of the superlinear elliptic problems − Δ u = a ( x ) u p on a bounded domain Ω in R N, where a ( x ) is Hölder continuous in Ω . Our main motivation is to study the case where a ( x ) ≥ 0, a ( x ) ≢ 0 and a ( x ) has some zero sets in Ω . We show that, in this case, the scaling arguments reduce the problem of a priori bounds to the Liouville-type results for the equation − Δ u = A ( x ′ ) u p in R N, where A is the continuous function defined on the subspace R k with 1 ≤ k ≤ N and x ′ ∈ R k . We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.