Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913430415343616 |
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| author | Gou, Tianxiang Radulescu, Vicentiu D. |
| author_facet | Gou, Tianxiang Radulescu, Vicentiu D. |
| contents | In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, \end{align*} where $n \geq 1$, $0<s<1$, $ω>-λ_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, and $λ_{1,s}>0$ is the lowest eigenvalue of the operator $(-Δ)^s + |x|^2$. This solves an open question raised in \cite{SS} concerning the uniqueness of solutions to the equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential Gou, Tianxiang Radulescu, Vicentiu D. Analysis of PDEs 35A02, 35R11 In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, \end{align*} where $n \geq 1$, $0<s<1$, $ω>-λ_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, and $λ_{1,s}>0$ is the lowest eigenvalue of the operator $(-Δ)^s + |x|^2$. This solves an open question raised in \cite{SS} concerning the uniqueness of solutions to the equation. |
| title | Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential |
| topic | Analysis of PDEs 35A02, 35R11 |
| url | https://arxiv.org/abs/2407.10126 |