Nontrivial Solutions for a (p, q)-Type Critical Choquard Equation on the Heisenberg Group

In this paper, we consider a critical (p, q) equation on the Heisenberg group of the following form: -ΔH,pu-ΔH,qu+V(ξ)(|u|p-2u+|u|q-2u)=μ∫HnF(ξ,u)|η-1ξ|λdξf(η,u)+|u|q∗-2u,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta _{H,p}u-\Delta _{H,q}u+V(\xi )(|u|^{p-2}u+|u|^{q-2}u)=\mu \int \limits _{{\mathbb {H}}^{n}} \frac{F(\xi ,u)}{|\eta ^{-1}\xi |^{\lambda }}{\text {d}}\xi f(\eta ,u)+|u|^{q^{*}-2}u, \end{aligned}$$\end{document}where the operator -ΔH,℘φ=divH(|DHφ|H℘-2DHφ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _{H,\varvec{\wp }}\varphi ={\text {div}}_H(|D_H\varphi |_H^{\varvec{\wp }-2}D_H\varphi )$$\end{document}, with ℘∈{p,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\wp }\in \{p,q\}$$\end{document}, is the proverbial horizontal ℘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\wp }$$\end{document}-Laplacian on the Heisenberg group, 1<p<(2Q-λ)2Qq<q<Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 1< p<\frac{(2Q-\lambda )}{2Q}q< q < Q $$\end{document}, q∗=qQ/(Q-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{*} = qQ/(Q-q)$$\end{document} is the critical exponent, and Q=2n+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q = 2n + 2 $$\end{document} is the homogeneous dimension of Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^{n}$$\end{document}, μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} are some real parameters. Under the appropriate assumptions of potential functions V and f, the existence of entire solutions to the above equation on the Heisenberg group is obtained by using the mountain pass theorem and the concentration compactness principle. The results presented here extend or complete recent papers and are new to critical equations involving (p, q)-Laplacian operators and convolution terms on Heisenberg group.