含Hardy位势的非线性Schrödinger-Poisson方程正规化解的多重性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (2): 442-453.

含Hardy位势的非线性Schrödinger-Poisson方程正规化解的多重性

杜梦雪,李方卉,王征平*()

武汉理工大学数学科学中心武汉 430070

收稿日期:2021-06-23 出版日期:2022-04-26 发布日期:2022-04-18
通讯作者: 王征平 E-mail:zpwang@whut.edu.cn
基金资助:
国家自然科学基金(11871386);国家自然科学基金(11931012)

Multiplicity of Normalized Solutions for Nonlinear Schrödinger-Poisson Equation with Hardy Potential

Mengxue Du,Fanghui Li,Zhengping Wang*()

Center of Mathematics, Wuhan University of Technology, Wuhan 430070

Received:2021-06-23 Online:2022-04-26 Published:2022-04-18
Contact: Zhengping Wang E-mail:zpwang@whut.edu.cn
Supported by:
the NSFC(11871386);the NSFC(11931012)

摘要/Abstract

摘要：

该文研究了含Hardy位势的非线性Schrödinger-Poisson方程正规化解的多重性问题. 首先利用喷泉定理的思想定义了一个极小极大值序列, 然后证明这些极小极大值是限制在约束集合上的能量泛函的临界值, 从而得到了方程正规化解的多重性, 推广了相关文献的结果.

关键词: Schrödinger-Poisson方程, Hardy位势, 正规化解, 多重性

Abstract:

In this paper, we concern the multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson equation with Hardy potential. By using some ideas of the fountain theorem, we define a sequence of minimax values and prove that these minimax values are critical values of the energy functional limited to a constraint set. Then we get the multiplicity of normalized solutions, which extends some related results in the literature.

Key words: Schrödinger-Poisson equation, Hardy potential, Normalized solution, Multiplicity

中图分类号:

O175.2

杜梦雪, 李方卉, 王征平. 含Hardy位势的非线性Schrödinger-Poisson方程正规化解的多重性[J]. 数学物理学报, 2022, 42(2): 442-453.

Mengxue Du, Fanghui Li, Zhengping Wang. Multiplicity of Normalized Solutions for Nonlinear Schrödinger-Poisson Equation with Hardy Potential[J]. Acta mathematica scientia,Series A, 2022, 42(2): 442-453.

参考文献 12

1	D'Aprile T , Mugnai D . Solitary waves for the nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proceedings of the Royal Society of Edinburgh, 2004, 134 (5): 893- 906 doi: 10.1017/S030821050000353X
2	Ruiz D . The Schrödinger-Poisson equation under the effect of a nonlinear local term. Journal of Functional Analysis, 2006, 237 (2): 655- 674 doi: 10.1016/j.jfa.2006.04.005
3	Ambrosetti A . On Schrödinger-Poisson systems. Milan Journal of Mathematics, 2008, 76 (1): 257- 274 doi: 10.1007/s00032-008-0094-z
4	Bellazzini J , Jeanjean L , Luo T J . Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. Proceedings of the London Mathematical Society, 2013, 107 (2): 303- 339 doi: 10.1112/plms/pds072
5	Li G B , Peng S J , Yan S S . Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system. Communications in Contemporary Mathematics, 2010, 12 (6): 1069- 1092 doi: 10.1142/S0219199710004068
6	Jiang Y S , Zhou H S . Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential. Science China Mathematics, 2014, 57 (6): 1163- 1174 doi: 10.1007/s11425-014-4790-6
7	Luo T J . Multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson-Slater equations. Journal of Mathematical Analysis and Applications, 2014, 416 (1): 195- 204 doi: 10.1016/j.jmaa.2014.02.038
8	Wang Z P , Zhou H S . Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $R^3$. Calculus of Variations and Partial Differential Equations, 2015, 52 (3): 927- 943
9	Zeng X Y , Zhang L . Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials. Journal of Mathematical Analysis and Applications, 2017, 452 (1): 47- 61 doi: 10.1016/j.jmaa.2017.02.053
10	Bartsch T , De Valeriola S . Normalized solutions of nonlinear Schrödinger equations. Archiv der Mathematik, 2013, 100 (1): 75- 83 doi: 10.1007/s00013-012-0468-x
11	Jeanjean L . Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Analysis, 1997, 28 (10): 1633- 1659 doi: 10.1016/S0362-546X(96)00021-1
12	Berestycki H , Lions P L . Nonlinear scalar field equations, Ⅱ Existence of infinitely many solutions. Archive for Rational Mechanics and Analysis, 1983, 82 (4): 347- 375 doi: 10.1007/BF00250556

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