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

In this article, we are study the following Dirichlet problem with Choquard type non linearity -Delta Hu = au + (integral(Omega) vertical bar u(eta)vertical bar(Q lambda)/vertical bar eta(-1)xi vertical bar(lambda) dn) vertical bar u vertical bar(Q lambda-2)u in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded subset of the Heisenberg group H-N, N is an element of N with C-2 boundary and Delta(H) is the Kohn Laplacian on the Heisenberg group H-N. Here, Q(lambda)(*), = 2AQ-lambda/Q-2, Q = 2N + 2 and a is a positive real parameter. We derive the BrezisNirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of alpha. (C) 2020 Elsevier Inc. All rights reserved.