数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 539-563.doi: 10.1007/s10473-023-0205-5

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THE EXISTENCE AND STABILITY OF NORMALIZED SOLUTIONS FOR A BI-HARMONIC NONLINEAR SCHRÖDINGER EQUATION WITH MIXED DISPERSION*

Tingjian Luo1, Shijun Zheng2, Shihui Zhu3,†   

  1. 1. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China;
    2. Department of Mathematical Sciences, Georgia Southern University, Statesboro 30460-8093, USA;
    3. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China
  • 收稿日期:2022-03-08 修回日期:2022-07-04 出版日期:2023-03-25 发布日期:2023-04-12
  • 通讯作者: †Shihui Zhu, E-mail: shihuizhumath@163.com; shihuizhumath@sicnu.edu.cn.
  • 作者简介:Tingjian Luo, E-mail: luotj@gzhu.edu.cn; Shijun Zheng, E-mail: szheng@GeorgiaSouthern.edu
  • 基金资助:
    Tingjian Luo was partially supported by the National Natural Science Foundation of China (11501137) and the Guangdong Basic and Applied Basic Research Foundation (2016A030310258, 2020A1515011019). Shihui Zhu was partially supported by the National Natural Science Foundation of China (11501395, 12071323).

THE EXISTENCE AND STABILITY OF NORMALIZED SOLUTIONS FOR A BI-HARMONIC NONLINEAR SCHRÖDINGER EQUATION WITH MIXED DISPERSION*

Tingjian Luo1, Shijun Zheng2, Shihui Zhu3,†   

  1. 1. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China;
    2. Department of Mathematical Sciences, Georgia Southern University, Statesboro 30460-8093, USA;
    3. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China
  • Received:2022-03-08 Revised:2022-07-04 Online:2023-03-25 Published:2023-04-12
  • Contact: †Shihui Zhu, E-mail: shihuizhumath@163.com; shihuizhumath@sicnu.edu.cn.
  • About author:Tingjian Luo, E-mail: luotj@gzhu.edu.cn; Shijun Zheng, E-mail: szheng@GeorgiaSouthern.edu
  • Supported by:
    Tingjian Luo was partially supported by the National Natural Science Foundation of China (11501137) and the Guangdong Basic and Applied Basic Research Foundation (2016A030310258, 2020A1515011019). Shihui Zhu was partially supported by the National Natural Science Foundation of China (11501395, 12071323).

摘要: In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr\"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denoting by $Q_p$ the ground state for the BNLS with $\mu=0$, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $\mu\in ( -\lambda_0, \infty)$ for some $\lambda_0=\lambda_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$, we prove the orbital stability on a certain mass level below $\|Q^*\|_{L^2}$, provided that $\mu\in (-\lambda_1,0)$, where $\lambda_1=\frac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac8d]$.

关键词: elliptic equations, bi-harmonic operator, normalized solutions, profile decomposition

Abstract: In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr\"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denoting by $Q_p$ the ground state for the BNLS with $\mu=0$, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $\mu\in ( -\lambda_0, \infty)$ for some $\lambda_0=\lambda_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$, we prove the orbital stability on a certain mass level below $\|Q^*\|_{L^2}$, provided that $\mu\in (-\lambda_1,0)$, where $\lambda_1=\frac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac8d]$.

Key words: elliptic equations, bi-harmonic operator, normalized solutions, profile decomposition