Uniqueness of Positive Solutions for Fractional Schrödinger Equations with General Nonlinearities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909537915633664 |
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| author | Li, Xinyu Song, Linjie |
| author_facet | Li, Xinyu Song, Linjie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of Positive Solutions for Fractional Schrödinger Equations with General Nonlinearities Li, Xinyu Song, Linjie Analysis of PDEs 35A02, 35B09, 35R11 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. |
| title | Uniqueness of Positive Solutions for Fractional Schrödinger Equations with General Nonlinearities |
| topic | Analysis of PDEs 35A02, 35B09, 35R11 |
| url | https://arxiv.org/abs/2401.02795 |