Non-degeneracy of solution for critical Lane-Emden systems with linear perturbation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866908704517914624 |
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| author | Guo, Yuxia Hu, Yichen Peng, Shaolong |
| author_facet | Guo, Yuxia Hu, Yichen Peng, Shaolong |
| contents | In this paper, we consider the following elliptic system \begin{equation*} \begin{cases} -Δu = |v|^{p-1}v +ε(αu + β_1 v), &\hbox{ in }Ω, \\-Δv = |u|^{q-1}u+ε(β_2 u +αv), &\hbox{ in }Ω, \\u=v=0,&\hbox{ on }\partialΩ, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, $ε$ is a small parameter, $α$, $ β_1$ and $ β_2$ are real numbers, $(p,q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation*} \begin{split} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{split} \end{equation*} We first revisited the blowing-up solutions constructed in \cite{Kim-Pis} and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06750 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-degeneracy of solution for critical Lane-Emden systems with linear perturbation Guo, Yuxia Hu, Yichen Peng, Shaolong Analysis of PDEs 35B33, 35J47, 35J67 In this paper, we consider the following elliptic system \begin{equation*} \begin{cases} -Δu = |v|^{p-1}v +ε(αu + β_1 v), &\hbox{ in }Ω, \\-Δv = |u|^{q-1}u+ε(β_2 u +αv), &\hbox{ in }Ω, \\u=v=0,&\hbox{ on }\partialΩ, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, $ε$ is a small parameter, $α$, $ β_1$ and $ β_2$ are real numbers, $(p,q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation*} \begin{split} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{split} \end{equation*} We first revisited the blowing-up solutions constructed in \cite{Kim-Pis} and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions. |
| title | Non-degeneracy of solution for critical Lane-Emden systems with linear perturbation |
| topic | Analysis of PDEs 35B33, 35J47, 35J67 |
| url | https://arxiv.org/abs/2210.06750 |