Multiplicity of solutions for the degenerate Kirchhoff system with critical nonlinearity on the Heisenberg group

Abstract

In this paper, we deal with the multiple solutions for the degenerate Kirchhoff system with critical nonlinearity on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle -M \Big (\int_{\Omega}|\nabla_Hu|^{2}d\xi\Big )\Delta_{H} u + \phi_{1}|u|^{q-2} u=\lambda|u|^{2} u+F_{u}(\xi, u, v) & \text { in } \Omega, \\[8pt] \displaystyle -M\Big (\int_{\Omega}|\nabla_Hv|^{2}d\xi\Big )\Delta_{H} v + \phi_{2}|v|^{q-2} v=\lambda|v|^{2} v+F_{v}(\xi, u, v)& \text { in } \Omega, \\ \displaystyle -\Delta_{H} \phi_{1}=|u|^q, \quad-\Delta_{H} \phi_{2}=|v|^q & \text { in } \Omega, \\ \displaystyle \phi_{1}=\phi_{2}=u = v =0& \text { on } \partial \Omega,\end{cases} \end{equation*} where $\Delta_{H}$ is the Kohn-Laplacian, $1 < q < 2$, ${\lambda}$ is a positive real parameter, and $F=F(\xi,u,v),F_{u}=\frac{\partial F}{\partial u}$, $F_{v}=\frac{\partial F}{\partial v}$. Under some suitable assumptions on the Kirchhoff function $M$ and $F$, together with the symmetric mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in the degenerate cases on the Heisenberg group. The result of this paper extends or else completes recent papers and is new in several directions for the critical Kirchhoff-Poisson systems on the Heisenberg group.