Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem. (June 2015)

• Record Type: Journal Article
• Authors: Xiang, Mingqi; Zhang, Binlin; Guo, Xiuying
• Is Part Of: Nonlinear analysis. Volume 120(2015)
• Page Start: 299
• Page End: 313
• Publication Date: 2015-06
• Subjects: 35R11 -- 35A15 -- 35J60; Integro-differential operators -- Fractional Laplacian -- Kirchhoff type equations -- Fountain Theorem -- Dual Fountain Theorem; 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.2015.03.015
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we use the Fountain Theorem and the Dual Fountain Theorem to study the existence of infinitely many solutions for Kirchhoff type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacian of Kirchhoff type: { M ( ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) ( − Δ ) s u ( x ) − λ u = f ( x, u ) in Ω u = 0 in R N ∖ Ω, where Ω is a smooth bounded domain of R N, ( − Δ ) s is the fractional Laplacian operator with 0 < s < 1 and 2 s < N, λ is a real parameter, M is a continuous and positive function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz type condition.