The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms. (September 2019)

• Record Type: Journal Article
• Authors: Fiscella, Alessio; Mishra, Pawan Kumar
• Is Part Of: Nonlinear analysis. Volume 186(2019)
• Page Start: 6
• Page End: 32
• Publication Date: 2019-09
• Subjects: 35J75 -- 35R11 -- 49J35; Kirchhoff type problems -- Fractional Laplacian -- Singularities -- Critical nonlinearities -- Nehari manifolds; 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.2018.09.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In the present paper, we study the following singular Kirchhoff problem M ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ( − Δ ) s u = λ f ( x ) u − γ + g ( x ) u 2 s ∗ − 1 in Ω, u > 0 in Ω, u = 0 in R N ∖ Ω, where Ω ⊂ R N is an open bounded domain, dimension N > 2 s with s ∈ ( 0, 1 ), 2 s ∗ = 2 N ∕ ( N − 2 s ) is the fractional critical Sobolev exponent, parameter λ > 0, exponent γ ∈ ( 0, 1 ), M models a Kirchhoff coefficient, f ∈ L 2 s ∗ 2 s ∗ + γ − 1 ( Ω ) is a positive weight, while g ∈ L ∞ ( Ω ) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ . This approach allows us to avoid any restriction on the boundary of Ω .