Existence results for Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity. (September 2020) - Song, Yueqiang; Zhao, Fu; Pu, Hongling; Shi, Shaoyun
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.