Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity. (September 2019)

• Record Type: Journal Article
• Authors: Goel, Divya; Sreenadh, K.
• Is Part Of: Nonlinear analysis. Volume 186(2019)
• Page Start: 162
• Page End: 186
• Publication Date: 2019-09
• Subjects: 35A15 -- 35J60 -- 35J20; Kirchhoff equation -- Hardy–Littlewood–Sobolev critical exponent -- Positive solution; 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.2019.01.035
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the following Kirchhoff–Choquard equation − M ( ‖ ∇ u ‖ L 2 2 ) Δ u = λ f ( x ) | u | q − 2 u + ∫ Ω | u ( y ) | 2 μ ∗ | x − y | μ d y | u | 2 μ ∗ − 2 u in Ω, u = 0 on ∂ Ω, where Ω is a bounded domain in R N ( N ≥ 3 ) with C 2 boundary, 2 μ ∗ = 2 N − μ N − 2, 1 < q ≤ 2, and f is a continuous real valued sign changing function. When 1 < q < 2, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also prove the existence of a positive solution when q = 2 using the Mountain Pass Lemma.