Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914252320669696 |
|---|---|
| author | Gou, Tianxiang |
| author_facet | Gou, Tianxiang |
| contents | In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$, $0<s<1$, $ω>-λ_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, $λ_{1,s}>0$ is the lowest eigenvalue of $(-Δ)^s + |x|^2$. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^n$, where $\mathcal{F}$ denotes the Fourier transform. This solves an open question in \cite{SS} concerning the uniqueness of ground states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_12068 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential Gou, Tianxiang Analysis of PDEs 35A02, 35R11 In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$, $0<s<1$, $ω>-λ_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, $λ_{1,s}>0$ is the lowest eigenvalue of $(-Δ)^s + |x|^2$. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^n$, where $\mathcal{F}$ denotes the Fourier transform. This solves an open question in \cite{SS} concerning the uniqueness of ground states. |
| title | Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential |
| topic | Analysis of PDEs 35A02, 35R11 |
| url | https://arxiv.org/abs/2208.12068 |