Kirchhoff-type problems involving the fractional p-Laplacian on the Heisenberg group

In this paper, we are interested in the existence of solutions for a class of Kirchhoff-type problems driven by a non-local integro-differential operator with the homogeneous Dirichlet boundary conditions on the Heisenberg group as follows: M(∬H2N|u(ξ)-u(η)|pK(η-1∘ξ)dξdη)£Kpu=f(ξ,u)inΩ,u=0inHN\Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} M(\iint _{{\mathbb {H}}^{2N}}|u(\xi )-u(\eta )|^{p}K(\eta ^{-1}\circ \xi )d\xi \,d\eta )\pounds ^{p}_{K}u=f(\xi ,u) &{} { \text{ in } } \Omega ,\\ u=0 &{} { \text{ in } } {\mathbb {H}}^N \setminus \Omega , \end{array} \right. \end{aligned}$$\end{document}where £Kp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pounds ^{p}_{K}$$\end{document} is a non-local integro-differential operator with singular kernel K,Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K,\Omega$$\end{document} is an open bounded subset of the Heisenberg group 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} with Lipshcitz boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega$$\end{document}. Under some suitable assumptions on the functions M and f, together with the variational methods and the mountain pass theorem, we discuss the existence of weak solutions for the above problem on the Heisenberg group.