On Existence and Stability Results for Normalized Ground States of Mass-Subcritical Biharmonic Nonlinear Schrödinger Equation on [math]

IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED SIAM Journal on Mathematical Analysis Pub Date : 2024-07-01 DOI:10.1137/22m1543707
Hichem Hajaiej, Yongming Luo, Lingjie Song
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Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4415-4439, August 2024.
Abstract. We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space [math]. Following the crucial scaling arguments introduced in [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number [math] that sharply determines the [math]-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] for determining the sharp [math]-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that [math]-independence of ground states with small mass still holds in the case [math] and [math]. Additionally, we also prove that ground states with sufficiently large mass must possess nontrivial [math]-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.
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论质量次临界双谐波非线性薛定谔方程在[数学]上的归一化基态的存在性和稳定性结果
SIAM 数学分析期刊》,第 56 卷第 4 期,第 4415-4439 页,2024 年 8 月。 摘要。我们研究了乘积空间[math]上的聚焦质量次临界双谐非线性薛定谔方程(BNLS)。根据[Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp.此外,在不存在低阶弥散的情况下,我们证明了临界质量数[math]的存在,该临界质量数极大地决定了推导出的基态的[math]依赖性。在混合色散情况下,我们遇到了一个重大挑战,因为 BNLS 不再是尺度不变的,而且[Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp.本文的主要创新之处在于解决了这一棘手而有趣的问题:通过使用不同的缩放论证,我们证明了在[math]和[math]情况下,小质量基态的[math]依赖性仍然成立。此外,我们还通过一些新颖的检验函数构造,证明了具有足够大质量的基态必须具有非简单的[数学]依赖性。后者尤其适用于处于全质量次临界机制的所有参数。
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来源期刊
CiteScore
3.30
自引率
5.00%
发文量
175
审稿时长
12 months
期刊介绍: SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.
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