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} -Δ_{$\mathbb{H}$}u=\left(\int_{$\mathbb{H}$^{n}}\frac{|u(η)|^{Q^{\ast}_μ}}{|η^{-1}ξ|^μ}$\mathrm{d}$η\right)|u|^{Q^{\ast}μ-2}u,~~~ξ,η\in$\mathbb{H}$^{n}, \end{equation} where ${\mathrm{\Delta }}_{\mathbb{H}}$ 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 $Q_{\mu }^{\ast }$ 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 $\epsilon \to 0$ \begin{equation}\label{0.2} \left{ \begin{aligned} &-Δ{$\mathbb{H}$}u=\varepsilon u+\left(\int_Ω\frac{|u(η)|^{Q^{\ast}_μ}}{|η^{-1}ξ|^μ}$\mathrm{d}$η\right)|u|^{Q^{\ast}_μ-2}u,&&$\mathrm{i}\mathrm{n}$~Ω\subset $\mathbb{H}$^{n}, &u=0,&&$\mathrm{o}\mathrm{n}$~\partialΩ. \end{aligned} \right. \end{equation}