含有Dirac位势的非线性S-P方程的约束极小元

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 907-913.

含有Dirac位势的非线性S-P方程的约束极小元

陈熙(),王征平^*()

武汉理工大学数学系武汉 430070

收稿日期:2023-10-31 修回日期:2024-01-02 出版日期:2024-08-26 发布日期:2024-07-26
通讯作者: *王征平, E-mail:actams@apm.ac.cn
作者简介:陈熙, E-mail:actams@wipm.ac.cn
基金资助:
国家自然科学基金(12371118);国家自然科学基金(11931012);国家自然科学基金(12071482)

Constrained Minimizers of Nonlinear S-P Equations with Dirac Potentials

Chen Xi(),Wang Zhengping^*()

Department of Mathematics, Wuhan University of Technology, Wuhan 430070

Received:2023-10-31 Revised:2024-01-02 Online:2024-08-26 Published:2024-07-26
Supported by:
NSFC(12371118);NSFC(11931012);NSFC(12071482)

摘要/Abstract

摘要：

该文研究了一类含有 Dirac 位势的非线性 Schrödinger-Poisson 方程的约束变分问题, 在一定的参数和指标假设条件下, 证明了约束极小元的存在性, 推广了文献 [2] 中的相关结论.

关键词: Dirac 位势, Schrödinger-Poisson 方程, 约束极小元

Abstract:

In this paper, we study the constrained variational problem of a class of nonlinear Schrödinger-Poisson equations with Dirac potentials. Under the assumptions of certain parameters and indices, we prove the existence of constrained minimizers, and the relevant conclusions is further extended in reference [2].

Key words: Dirac potentials, Schrödinger-Poisson equations, Constrained minimizers

中图分类号:

O177.91

陈熙, 王征平. 含有Dirac位势的非线性S-P方程的约束极小元[J]. 数学物理学报, 2024, 44(4): 907-913.

Chen Xi, Wang Zhengping. Constrained Minimizers of Nonlinear S-P Equations with Dirac Potentials[J]. Acta mathematica scientia,Series A, 2024, 44(4): 907-913.

参考文献 18

[1]	Abramowitz M, Stegun I A, Romer R H. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: US Govt Print, 2006
[2]	Adami R, Boni F, Carlone R, et al. Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect. Journal of Mathematical Physics, 2022, 63(7): 071501
[3]	Adami R, Boni F, Carlone R, et al. Ground states for the planar NLSE with a point defect as minimizers of the constrained energy. Calculus of Variations and Partial Differential Equations, 2022, 61(5): Article 195
[4]	Albeverio S, Gesztesy F, Hoegh-Krohn R, et al. Solvable Models in Quantum Mechanics. Berlin: Springer Science and Business Media, 2012
[5]	Bellazzini J, Siciliano G. Scaling properties of functionals and existence of constrained minimizers. Journal of Functional Analysis, 2011, 261(9): 2486-2507
[6]	Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proceedings of the American Mathematical Society, 1983, 88( 3): 486-490
[7]	Brézis H, Lions P L. A note on isolated singularities for linear elliptic equations. Journal of Mathematical Analysis and Applications, 1981, 7: 263-266
[8]	Cazenave T. Semilinear Schrödinger Equations. Providence, RI: Amer Math Soc, 2003
[9]	Finco D, Noja D. Blow-up and instability of standing waves for the NLS with a point interaction in dimension two. Z Angew Math Phys, 2023, 74: Article 162
[10]	Fukaya N, Georgiev V, Ikeda M. On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction. Journal of Differential Equations, 2022, 321: 258-295
[11]	Georgiev V, Michelangeli A, Scandone R. Standing waves and global well-posedness for the 2d Hartree equation with a point interaction. Comm Part Differ Equa, 2024, 49(3):242-278
[12]	Ghergu M, Kim S, Shahgholian H. Isolated singularities for semilinear elliptic systems with power-law nonlinearity. Analysis and Partial Differential Equations, 2020, 13(3): 701-739
[13]	Gidas B, Ni W M, Nirenberg L. Symmetry and related properties via the maximum principle. Communications in Mathematical Physics, 1979, 68(3): 209-243
[14]	Le Coz S, Fukuizumi R, Fibich G, et al. Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Physica D: Nonlinear Phenomena, 2008, 237(8): 1103-1128
[15]	Lions P L. The concentration-compactness principle in the calculus of variations. The Locally compact case, part 2. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 1984, 1(4): 223-283
[16]	Ni W M. Existence and nonexistence theorems for ground states of quasilinear partial differential equations: The anomalous case. Accademia Nazionale dei Lincei, 1986, 77: 231-257
[17]	Ni W M, Serrin J. Nonexistence theorems for singular solutions of quasilinear partial differential equations. Communications on Pure and Applied Mathematics, 1986, 39(3): 379-399
[18]	Shamriz E, Chen Z, Malomed B A, et al. Singular mean-field states: A brief review of recent results. Condensed Matter, 2020, 5(1): 20