On a nonlinear eigenvalue problem for generalized Laplacian in Orlicz–Sobolev spaces. (January 2020)

• Record Type: Journal Article
• Authors: Youssfi, Ahmed; Khatri, Mohamed Mahmoud Ould
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 46E30 -- 35P30 -- 35D30; Orlicz–Sobolev spaces -- Nonlinear eigenvalue problems -- Harnack inequality; 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.2019.111607
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the p -Laplacian. The natural framework in which we consider such equations is that of Orlicz–Sobolev spaces. We exhibit two positive constants λ 0 and λ 1 with λ 0 ≤ λ 1 such that λ 1 is an eigenvalue of the problem while any value λ < λ 0 cannot be so. By means of Harnack-type inequalities and a strong maximum principle, we prove the isolation of λ 1 on the right side. We emphasize that throughout the paper no Δ 2 -condition is needed.