On the critical Choquard-Kirchhoff problem on the Heisenberg group

Abstract

Abstract In this paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: M ( ‖ u ‖ 2 ) ( − Δ H u + V ( ξ ) u ) = ∫ H N ∣ u ( η ) ∣ Q λ ∗ ∣ η − 1 ξ ∣ λ d η ∣ u ∣ Q λ ∗ − 2 u + μ f ( ξ , u ) , M\left(\Vert u{\Vert }^{2})\left(-{\Delta }_{{\mathbb{H}}}u\left+V\left(\xi )u)=\left(\mathop{\int }\limits_{{{\mathbb{H}}}^{N}}\frac{| u\left(\eta ){| }^{{Q}_{\lambda }^{\ast }}}{| {\eta }^{-1}\xi {| }^{\lambda }}{\rm{d}}\eta \right)| u{| }^{{Q}_{\lambda }^{\ast }-2}u+\mu f\left(\xi ,u), where M M is the Kirchhoff function, Δ H {\Delta }_{{\mathbb{H}}} is the Kohn Laplacian on the Heisenberg group H N {{\mathbb{H}}}^{N} , f f is a Carathéodory function, μ > 0 \mu \gt 0 is a parameter and Q λ ∗ = 2 Q − λ Q − 2 {Q}_{\lambda }^{\ast }=\frac{2Q-\lambda }{Q-2} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We first establish a new version of the concentration-compactness principle for the Choquard equation on the Heisenberg group. Then, combining with the mountain pass theorem, we obtain the existence of nontrivial solutions to the aforementioned problem in the case of nondegenerate and degenerate cases.