Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications

We study the uniqueness and nondegeneracy of positive bubble solutions for the generalized energy-critical Hartree equation on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{0.1} -\Delta_{\mathbb{H}}u=\left(\int_{\mathbb{H}^{n}}\frac{|u(\eta)|^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}\mathrm{d}\eta\right)|u|^{Q^{\ast}_{\mu}-2}u,~~~\xi,\eta\in\mathbb{H}^{n}, \end{equation} where $\Delta_{\mathbb{H}}$ represents the Kohn Laplacian, $u(\eta)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$, $\mu\in (0,Q)$ is a real parameter and $Q^{\ast}_{\mu}$ is the upper critical exponent following the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By applying the Cayley transform, the spherical harmonic decomposition and the Funk-Hecke formula of the spherical harmonic function, we prove the nondegeneracy of positive bubble solutions for (\ref{0.1}). As an applications, we investigate the asymptotic behavior of the solutions for the Brezis-Nirenberg type problem as $\varepsilon\rightarrow 0$ \begin{equation}\label{0.2} \left\{ \begin{aligned} &-\Delta_{\mathbb{H}}u=\varepsilon u+\left(\int_{\Omega}\frac{|u(\eta)|^{Q^{\ast}_{\mu}}}{|\eta^{-1}\xi|^{\mu}}\mathrm{d}\eta \right)|u|^{Q^{\ast}_{\mu}-2}u,~~&&\mathrm{in}~\Omega\subset \mathbb{H}^{n}, &u=0,~~&&\mathrm{on}~\partial\Omega. \end{aligned} \right. \end{equation}