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具有一般非线性项的分数阶 Schrödinger 方程正解的唯一性

Uniqueness of Positive Solutions for Fractional Schrödinger Equations with General Nonlinearities

Xinyu Li, Linjie Song

备注：
There is an error in the proof of Lemma 3.2
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AI总结：

针对 Frank、Lenzmann 与 Silvestre 提出的开放问题，本文证明一大类凸非线性项下分数阶 Schrödinger 方程正解唯一。

摘要：

2013年，R.L. Frank 和 E. Lenzmann［R.L. Frank, E. Lenzmann，《分数阶 Laplace 算子在 $\mathbb{R}$ 中的非线性基态唯一性》，Acta Math. 210 (2) (2013) 261-318］研究了如下问题：$(-\Delta)^s u+u=u^{p-1}$，于 $\mathbb{R}^N$ 中，其中 $s\in(0,1)$、$N=1$、$p\in(2,2_s^*)$，而 $2_s^*$ 是临界分数阶 Sobolev 指数。他们证明了基态在相差平移的意义下是唯一的。随后在2016年，他们与 L. Silvestre 合作［R.L. Frank, E. Lenzmann, L. Silvestre，《分数阶 Laplace 算子径向解的唯一性》，Comm. Pure Appl. Math. 69 (9) (2016) 1671-1726］对高维情形（$N\geqslant2$）证明了类似的唯一性结果；在该研究中，他们提出了一个具有挑战性的开放问题，即将关于基态的非退化性和唯一性的结果推广到纯幂情形之外的非线性项 $f(u)$。据我们所知，这一问题至今仍未解决。本文旨在针对一大类凸非线性项，对这一开放问题给出完整的肯定回答。事实上，我们证明了正解的唯一性。

英文摘要：

In 2013, R.L. Frank and E. Lenzmann [R.L. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math. 210 (2) (2013) 261-318] study the following problem: \begin{align*} (-Δ)^su + u = u^{p-1} \quad \text{in } \mathbb{R}^N, \end{align*} where $s \in (0,1)$, $N = 1$, $p \in (2,2_s^*)$, and $2_s^*$ is the critical fractional Sobolev exponent. They proved that the ground state is unique (up to translations). Then in 2016, they, together with L. Silvestre [R.L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (9) (2016) 1671-1726] showed similar uniqueness results for high dimensions ($N \geqslant 2$), in which they proposed a challenging open problem to extend their results about non-degeneracy and uniqueness of ground states to nonlinearities $f(u)$ beyond the pure-power case. To the best of our knowledge, this question is still unresolved so far. In this paper, we aim to give a full affirmative answer to this open issue for a large class of convex nonlinearities. In fact, we prove the uniqueness of the positive solution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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