Uniqueness of bound states to $Δu-u+|u|^{p-1}u= 0$ in $\mathbb{R}^n$, $n\ge 3$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918153892659200 |
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| author | Tang, Moxun |
| author_facet | Tang, Moxun |
| contents | We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to $Δu +f(u)=0$ in $\mathbb{R}^n$, $u\in H^1(\mathbb{R}^n)$, $u\not\equiv 0$, $n\ge 3$. For the model nonlinearity $f(u)=-u+|u|^{p-1}u$, $1<p<(n+2)/(n-2)$, arisen from finding standing waves of Klein-Gordon equation or nonlinear Schrödinger equation, we show that, for each integer $k\ge 1$, the problem has a unique solution $u=u(|x|)$, $x\in \mathbb{R}^n$, up to translation and reflection, that has precisely $k$ zeros for $|x|>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of bound states to $Δu-u+|u|^{p-1}u= 0$ in $\mathbb{R}^n$, $n\ge 3$ Tang, Moxun Analysis of PDEs We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to $Δu +f(u)=0$ in $\mathbb{R}^n$, $u\in H^1(\mathbb{R}^n)$, $u\not\equiv 0$, $n\ge 3$. For the model nonlinearity $f(u)=-u+|u|^{p-1}u$, $1<p<(n+2)/(n-2)$, arisen from finding standing waves of Klein-Gordon equation or nonlinear Schrödinger equation, we show that, for each integer $k\ge 1$, the problem has a unique solution $u=u(|x|)$, $x\in \mathbb{R}^n$, up to translation and reflection, that has precisely $k$ zeros for $|x|>0$. |
| title | Uniqueness of bound states to $Δu-u+|u|^{p-1}u= 0$ in $\mathbb{R}^n$, $n\ge 3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.06915 |