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

TL;DR

This paper proves the nondegeneracy of positive solutions for a critical Hartree equation on the Heisenberg group and explores their asymptotic behavior in related boundary value problems.

Contribution

It establishes the nondegeneracy of positive bubble solutions for the critical Hartree equation on the Heisenberg group using harmonic analysis techniques.

Findings

Proved nondegeneracy of positive solutions.

Analyzed asymptotic behavior in Brezis-Nirenberg type problems.

Applied harmonic analysis tools like Cayley transform and Funk-Hecke formula.

Abstract

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}