Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908484335828992 |
|---|---|
| author | Yang, Minbo Zhang, Shuijin |
| author_facet | Yang, Minbo Zhang, Shuijin |
| contents | 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 $Δ_{\mathbb{H}}$ represents the Kohn Laplacian, $u(η)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$, $μ\in (0,Q)$ is a real parameter and $Q^{\ast}_μ$ 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}
&-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{in}~Ω\subset \mathbb{H}^{n},
&u=0,~~&&\mathrm{on}~\partialΩ.
\end{aligned}
\right. \end{equation} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications Yang, Minbo Zhang, Shuijin Analysis of PDEs 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 $Δ_{\mathbb{H}}$ represents the Kohn Laplacian, $u(η)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$, $μ\in (0,Q)$ is a real parameter and $Q^{\ast}_μ$ 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} &-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{in}~Ω\subset \mathbb{H}^{n}, &u=0,~~&&\mathrm{on}~\partialΩ. \end{aligned} \right. \end{equation} |
| title | Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.07719 |