On the critical Choquard-Kirchhoff problem on the Heisenberg group

Abstract

In this paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: M(‖u‖2)(−ΔHu+V(ξ)u)=∫HN∣u(η)∣Qλ∗∣η−1ξ∣λdη∣u∣Qλ∗−2u+μf(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 MM is the Kirchhoff function, ΔH{\Delta }_{{\mathbb{H}}} is the Kohn Laplacian on the Heisenberg group HN{{\mathbb{H}}}^{N}, ff is a Carathéodory function, μ>0\mu \gt 0 is a parameter and Qλ∗=2Q−λQ−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.

Keywords

• heisenberg group
• choquard-kirchhoff equation
• hardy-littlewood-sobolev critical exponent
• concentration-compactness principle
• variational methods
• 35j20
• 35r03
• 46e35