Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity

Abstract

We consider the following Kirchhoff–Choquard equation −M(‖∇u‖L22)Δu=λf(x)|u|q−2u+∫Ω|u(y)|2μ∗|x−y|μdy|u|2μ∗−2uinΩ,u=0on∂Ω, where Ω is a bounded domain in RN(N≥3) with C2 boundary, 2μ∗=2N−μN−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.