Existence results for Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity. (September 2020)

• Record Type: Journal Article
• Authors: Song, Yueqiang; Zhao, Fu; Pu, Hongling; Shi, Shaoyun
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: 35A15 -- 35J60 -- 35J20; Kirchhoff problem -- Hardy–Littlewood–Sobolev critical exponent -- Variational methods -- Concentration–compactness principle; 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.2020.111900
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study a class of Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity in R N . More precisely, we consider − a + b ∫ R N | ∇ u | 2 d x Δ u − a [ Δ ( u 2 ) ] u = λ I μ ∗ | u | p | u | p − 2 u + I μ ∗ | u | 2 2 μ ∗ | u | 2 2 μ ∗ − 2 u, where a > 0, b ≥ 0, N ≥ 3, 0 < μ < 4 N 3 N + 4, 2 ( N + μ ) N ≤ p < 2 μ ∗ ≔ 2 ( N − μ ) N − 2, λ > 0 and I μ is a Riesz potential. We prove the existence and multiplicity of solutions for the equation by variational methods together with concentration–compactness principle.