Standing wave solutions of a quasilinear Schrödinger equation in the small frequency limit
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| Formato: | Preprint |
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2024
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| author | Genoud, François Nodari, Simona Rota |
| author_facet | Genoud, François Nodari, Simona Rota |
| contents | This article is concerned with the quasilinear Schrödinger equation \[ Δu-ωu+|u|^{p-1}u+δΔ(|u|^2)u=0, \] where $δ>0$, $N=2$ and $p>1$ or $N\ge3$ and $1<p<\frac{3N+2}{N-2}$. After proving uniqueness and non-degeneracy of the positive solution $u_ω$ for all $ω>0$, our main results establish the asymptotic behavior of $u_ω$ in the limit $ω\to 0^+$. Three different regimes arise, termed 'subcritical', 'critical' and 'supercritical', corresponding respectively (when $N\ge3$) to $1<p<\frac{N+2}{N-2}$, $p=\frac{N+2}{N-2}$ and $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}$. In each case a limit equation is exhibited which governs, in a suitable scaling, the behavior of $u_ω$ in the limit $ω\to 0^+$. The critical case is the most challenging, technically speaking. In this case, the limit equation is the famous Lane-Emden-Fowler equation. A substantial part of our efforts is dedicated to the study of the function $ω\mapsto M(ω)=\int_{\mathbb{R}^N} u_ω^2$. We find that, for small $ω>0$, $M(ω)$ is increasing if $1<p\le 1+\frac4N$ and decreasing if $1+\frac4N< p\le\frac{N+2}{N-2}$. In the supercritical case, the monotonicity of $M(ω)$ depends on the dimension, except in the regime $p\ge 3+\frac4N$, where $M(ω)$ is always decreasing close to $ω=0$. The crucial role played by $M(ω)$ for the orbital stability of the standing wave $e^{iωt}u_ω$, and for the uniqueness of normalized ground states, is discussed in the introduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Standing wave solutions of a quasilinear Schrödinger equation in the small frequency limit Genoud, François Nodari, Simona Rota Analysis of PDEs 35Q55, 35B35 This article is concerned with the quasilinear Schrödinger equation \[ Δu-ωu+|u|^{p-1}u+δΔ(|u|^2)u=0, \] where $δ>0$, $N=2$ and $p>1$ or $N\ge3$ and $1<p<\frac{3N+2}{N-2}$. After proving uniqueness and non-degeneracy of the positive solution $u_ω$ for all $ω>0$, our main results establish the asymptotic behavior of $u_ω$ in the limit $ω\to 0^+$. Three different regimes arise, termed 'subcritical', 'critical' and 'supercritical', corresponding respectively (when $N\ge3$) to $1<p<\frac{N+2}{N-2}$, $p=\frac{N+2}{N-2}$ and $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}$. In each case a limit equation is exhibited which governs, in a suitable scaling, the behavior of $u_ω$ in the limit $ω\to 0^+$. The critical case is the most challenging, technically speaking. In this case, the limit equation is the famous Lane-Emden-Fowler equation. A substantial part of our efforts is dedicated to the study of the function $ω\mapsto M(ω)=\int_{\mathbb{R}^N} u_ω^2$. We find that, for small $ω>0$, $M(ω)$ is increasing if $1<p\le 1+\frac4N$ and decreasing if $1+\frac4N< p\le\frac{N+2}{N-2}$. In the supercritical case, the monotonicity of $M(ω)$ depends on the dimension, except in the regime $p\ge 3+\frac4N$, where $M(ω)$ is always decreasing close to $ω=0$. The crucial role played by $M(ω)$ for the orbital stability of the standing wave $e^{iωt}u_ω$, and for the uniqueness of normalized ground states, is discussed in the introduction. |
| title | Standing wave solutions of a quasilinear Schrödinger equation in the small frequency limit |
| topic | Analysis of PDEs 35Q55, 35B35 |
| url | https://arxiv.org/abs/2407.16179 |