Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity. (September 2019) - Goel, Divya; Sreenadh, K.
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.