A Brezis–Nirenberg splitting approach for nonlocal fractional equations. (June 2015)

• Record Type: Journal Article
• Authors: Molica Bisci, Giovanni; Servadei, Raffaella
• Is Part Of: Nonlinear analysis. Volume 119(2015)
• Page Start: 341
• Page End: 353
• Publication Date: 2015-06
• Subjects: primary 49J35 35A15 35S15 -- secondary 47G20 45G05; Fractional Laplacian -- Nonlocal problems -- Variational methods -- Critical point theory -- Integrodifferential operators; 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.2014.10.025
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we consider problems modeled by the following nonlocal fractional equation { ( − Δ ) s u + a ( x ) u = μ f ( u ) in Ω u = 0 in R n ∖ Ω, where s ∈ ( 0, 1 ) is fixed, Ω is an open bounded subset of R n, n > 2 s, with Lipschitz boundary, ( − Δ ) s is the fractional Laplace operator and μ is a real parameter. Under two different types of conditions on the functions a and f, by using a famous critical point theorem in the presence of splitting established by Brezis and Nirenberg, we obtain the existence of at least two nontrivial weak solutions for our problem.