Existence results for a class of critical fractional elliptic system on Heisenberg group

Abstract

In this paper, we study the existence of nontrivial solutions tothe critical fractional elliptic system on the Heisenberg group:$$\begin{gathered}\begin{cases}(-\Delta_{\mathbb{H}^{n}})^{s} u+\gamma u=\lambda f(\xi)|u|^{r-2} u+\frac{2 \alpha}{\alpha+\beta}|u|^{\alpha-2} u|v|^{\beta} & \text { in } \Omega, \\(-\Delta_{\mathbb{H}^{n}})^{s} v+\nu v=\mu g(\xi)|v|^{r-2} v+\frac{2 \beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2} v & \text { in } \Omega, \\u = v =0 & \text { in } \mathbb{H}^{n}\setminus \Omega,\end{cases}\end{gathered}$$where ${\Delta_{\mathbb{H}^{n}}}$ is the Kohn-Laplacian, $s\in(0,1)$, $ Q > 2 s$ and $\Omega\subset\mathbb{H}^{n}$ is a boundeddomain with smooth boundary. $\lambda, \mu,\gamma,\nu$ are positive real parameters and $ Q=2n+2$ is the homogeneousdimension of ${\mathbb{H}^{n}}$. The exponent $r$ satisfies$1 < r < 2^{*}_{s}$, $\alpha > 1, \beta > 1$ satisfy$2 < \alpha+\beta=2_{s}^{*}$, $2_{s}^{*}= 2Q/(Q-2s)$ being thefractional critical Sobolev exponent. The results presented hereextend or complete recent papers and are new to critical fractionalelliptic system on Heisenberg group.