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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.04139 |
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| _version_ | 1866913532457517056 |
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| author | Li, Xuemei Liu, Chenxi Tang, Xingdong Xu, Guixiang |
| author_facet | Li, Xuemei Liu, Chenxi Tang, Xingdong Xu, Guixiang |
| contents | In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*}
-{Δu}\sts{x}
-{\bmα}\sts{N,λ}
\int_{\R^N}
{ \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}λ} }\diff{y}\,
u^{p-1}\sts{x}
=0,\quad x\in \R^N \end{equation*} where $N\geq 3$, $0<λ<N$, $p=\frac{2N-λ}{N-2}$ and ${\bmα}\sts{N,λ}$ is a normalized constant such that $ u(x)=\left(1+|x|^2\right)^{-\frac{N-2}{2} }$ is a bubble solution of the equation \eqref{NLH}. It solves an open nondegeneracy problem in \cite{MWX:Hartree, GMYZ2022cvpde} and generalizes the partial nondegeneracy results in \cite{DY2019dcds, GWY2020na, LTX2021} to the full range $0<λ<N$. The key observation is that by use of the stereographic projection $\mathcal{S}$, the weighted pushforward map $\mathcal{S}_*$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\mathcal{H}_1^{N+1}$ of degree one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04139 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations Li, Xuemei Liu, Chenxi Tang, Xingdong Xu, Guixiang Analysis of PDEs Primary 35B09, 47A74. Secondly 35P10, 42B37 In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{Δu}\sts{x} -{\bmα}\sts{N,λ} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}λ} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $N\geq 3$, $0<λ<N$, $p=\frac{2N-λ}{N-2}$ and ${\bmα}\sts{N,λ}$ is a normalized constant such that $ u(x)=\left(1+|x|^2\right)^{-\frac{N-2}{2} }$ is a bubble solution of the equation \eqref{NLH}. It solves an open nondegeneracy problem in \cite{MWX:Hartree, GMYZ2022cvpde} and generalizes the partial nondegeneracy results in \cite{DY2019dcds, GWY2020na, LTX2021} to the full range $0<λ<N$. The key observation is that by use of the stereographic projection $\mathcal{S}$, the weighted pushforward map $\mathcal{S}_*$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\mathcal{H}_1^{N+1}$ of degree one. |
| title | Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations |
| topic | Analysis of PDEs Primary 35B09, 47A74. Secondly 35P10, 42B37 |
| url | https://arxiv.org/abs/2304.04139 |