四阶薛定谔方程归一化解存在性的另一个证明——献给李工宝教授 70 寿辰

Acta mathematica scientia,Series A ›› 2025, Vol. 45 ›› Issue (6): 1928-1941.

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Another Proof on the Existence of Normalized Solution to a Fourth-Order Schrödinger Equation

Mengxia Wu¹, Tao Yang¹^,^*(), Hua Zhang²

¹School of Mathematical Sciences, Zhejiang Normal University, Zhejiang Jinhua 321004
²Yunxi No.1 Senior High School, Hubei Shiyan 442600

Received:2025-05-14 Revised:2025-07-18 Online:2025-12-26 Published:2025-11-18
Contact: Tao Yang E-mail:yangtao@zjnu.edu.cn
Supported by:
NSFC(12201564);Scientific Research Fund of Zhejiang Normal University(YS304221948)

Abstract

Abstract:

In this paper, we consider the existence and asymptotic properties of normalized solutions to a fourth-order Schrödinger equation with a positive second-order dispersion coefficient. In the mass supercritical regime, we study two types of local minimization problems and prove their equivalence in order to avoid the dependence of mass with respect to the locally constraint radius. Then, we prove the compactness of the corresponding minimizing sequences and the existence of ground states. Furthermore, by utilizing subtle energy estimates and analysis, we derive the asymptotic behavior of the ground state and the Lagrange multiplier as the parameter vanishes. This paper removes the radial symmetry condition in (Sci China Math, 2023, 66: 1237--1262), and provides an alternative but more transparent proof than that of (J Differential Equations, 2022, 330: 1--65).

Key words: fourth-order Schr?dinger equation, normalized solution, local minimization method.

CLC Number:

O175.23

Mengxia Wu, Tao Yang, Hua Zhang. Another Proof on the Existence of Normalized Solution to a Fourth-Order Schrödinger Equation[J].Acta mathematica scientia,Series A, 2025, 45(6): 1928-1941.

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References 15

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Another Proof on the Existence of Normalized Solution to a Fourth-Order Schrödinger Equation

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