Existence and nonexistence results for Kohn Laplacian with Hardy-Littlewood-Sobolev critical exponents

Abstract

In this article, we are study the following Dirichlet problem with Choquard type non linearity−ΔHu=au+(∫Ω|u(η)|Qλ⁎|η−1ξ|λdη)|u|Qλ⁎−2uinΩ,u=0 on ∂Ω, where Ω is a smooth bounded subset of the Heisenberg group HN,N∈N with C2 boundary and ΔH is the Kohn Laplacian on the Heisenberg group HN. Here, Qλ⁎=2Q−λQ−2,Q=2N+2 and a is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of a.