Some existence results for critical nonlocal Choquard equation on the Heisenberg group

Abstract

This paper deals with the following critical nonlocal Choquard equation on the Heisenberg group: −(a−b∫Ω|∇Hu|2dξ)ΔHu=μ|u|q−2u+∫Ω|u(η)|Qλ∗|η−1ξ|λdη|u|Qλ∗−2uinΩ,u=0on∂Ω,where Ω⊂HN is a smooth bounded domain, ΔH is the Kohn-Laplacian on the Heisenberg group HN, 1<q<2 or 2<q<Qλ∗, a,b>0, μ>0, 0<λ<4, and Qλ∗=2Q−λQ−2 is the critical exponent. Existence results are obtained by using the Ekeland variational principle, Clark critical point theorem, mountain pass theorem, and Krasnoselskii genus theorem, respectively. Due to critical nonlinearities as well as the presence of the double non-local teams, there are some difficulties on the Heisenberg group’s framework. Our results are new even in the Euclidean case.