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Instability of standing waves for fractional NLS with combined nonlinearities

Part of: Equations of mathematical physics and other areas of application Partial differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  27 November 2024

Zaizheng Li ,
Haijun Luo  and
Zhitao Zhang
Zaizheng Li
Affiliation:
School of Mathematical Sciences, Hebei Center for Applied Mathematics, Hebei Normal University, Shijiazhuang 050024, Hebei, China (zaizhengli@hebtu.edu.cn)
Haijun Luo
Affiliation:
School of Mathematics, Hunan Provincial Key Laboratory of Intelligent Information, Processing and Applied Mathematics, Hunan University, Changsha 410082, Hunan, China (luohj@hnu.edu.cn) (corresponding author)
Zhitao Zhang
Affiliation:
HLM, Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China (zzt@math.ac.cn)
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Abstract

We study the existence and strong instability of standing wave solutions for the fractional nonlinear Schrödinger equation

\begin{equation*}\begin{cases}\mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\\\psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N,\end{cases}\end{equation*}

where $N\geq2$, $0 \lt s \lt 1$, $2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$, and $\mu\in\mathbb{R}$. The primary challenge lies in the inhomogeneity of the nonlinearity.We deal with the following three cases: (i) for $2 \lt q \lt p \lt 2+4s/N$ and µ < 0, there exists a threshold mass a0 for the existence of the least energy normalized solution; (ii) for $2+4s/N \lt q \lt p \lt 2_s^*$ and µ > 0, we reveal the existence of the ground state solution, explore the strong instability of standing waves, and provide a blow-up criterion; (iii) for $2 \lt q\leq2+4s/N \lt p \lt 2_s^*$ and µ < 0, the strong instability of standing wave solutions is demonstrated. These findings are illuminated through variational characterizations, the profile decomposition, and the virial estimate.

Keywords

fractional NLSinstabilitystanding wavevariational characterizationground state solution

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 35B35: Stability 35Q55: NLS-like equations (nonlinear Schrödinger)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 29
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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