临界情形下Schrödinger-Maxwell方程的基态解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 475-483.

临界情形下Schrödinger-Maxwell方程的基态解

方立婉¹(),黄文念²(),汪敏庆²()

¹ 广西科技师范学院数学与计算机科学学院广西来宾 546199
² 广西师范大学数学与统计学院广西桂林 541004

收稿日期:2017-04-28 出版日期:2019-06-26 发布日期:2019-06-27
作者简介:方立婉, fangliwan1992@163.com|黄文念, csuhuangwn@163.com|汪敏庆, hgncwangmq@126.com
基金资助:
广西师范大学科学研究基金(2014ZD001);广西自然科学基金(2015GXNSFBA139018);2017广西研究生教育创新计划项目(XYCZ2017074)

Ground-State Solutions for Schrödinger-Maxwell Equations in the Critical Growth

Liwan Fang¹(),Wennian Huang²(),Minqing Wang²()

¹ School of Mathematics and Computer Science, Guangxi Science and Technology Normal University; Guangxi Laibin 546199
² School of Mathematics and Statistics, Guangxi Normal University, Guangxi Guilin 541004

Received:2017-04-28 Online:2019-06-26 Published:2019-06-27
Supported by:
the Science Research Fund of Guangxi Normal University(2014ZD001);the Natural Science Foundation of Guangxi(2015GXNSFBA139018);the Postgraduate Education Innovation Plan Project of Guangxi in 2017(XYCZ2017074)

摘要/Abstract

摘要：

该文主要研究下面的Schrödinger-Maxwell方程

$\left\{\begin{array}{ll} -\triangle u+V(x)u-(K(x)+\alpha)\phi u=\beta|u|^{4}u+b(x)|u|^{p-1}u, &(x,u)\in(\mathbb{R}^{3},\mathbb{R}),\\ \triangle\phi=(K(x)+\alpha)u^{2},& (x,u)\in(\mathbb{R}^{3},\mathbb{R}) \end{array} \right.$

基态解的存在性，其中β是正常数.当V和K以及b（x）满足某些假设条件时，运用变分法和临界点理论，可以证明当α < 0和p∈（3，4）时，上面的方程至少存在一个基态解.

关键词: Schrödinger-Maxwell方程, 临界点理论, 临界情形, 基态解, Nehari流形

Abstract:

In this paper, we study the existence of the ground state solutions for the following Schrödinger-Maxwell equations

where β is a positive constant. Under some assumptions on V, K and b(x), by using the variational method and critical point theorem, we prove that such a class of equations has at least a ground state solution for α < 0 and p ∈ (3, 4).

Key words: Schrödinger-Maxwell equations, Critical point theorem, Critical growth, Ground state solution, Nehari manifold

中图分类号:

O175.25

方立婉,黄文念,汪敏庆. 临界情形下Schrödinger-Maxwell方程的基态解[J]. 数学物理学报, 2019, 39(3): 475-483.

Liwan Fang,Wennian Huang,Minqing Wang. Ground-State Solutions for Schrödinger-Maxwell Equations in the Critical Growth[J]. Acta mathematica scientia,Series A, 2019, 39(3): 475-483.

参考文献 17

1	Zhang J . On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth. Nonlinear Analysis: Theory, Methods and Applications, 2012, 75: 6391- 6401 doi: 10.1016/j.na.2012.07.008
2	Liu Z S , Guo S J . On ground state solutions for the Schrödinger-Poisson equations with critical growth. Journal of Mathematical Analysis and Applications, 2014, 412: 435- 448 doi: 10.1016/j.jmaa.2013.10.066
3	Yang M H , Han Z Q . Existence and multiplicity results for the nonlinear Schrödinger-Poisson systems. Nonlinear Analysis: Real World Applications, 2012, 13: 1093- 1101 doi: 10.1016/j.nonrwa.2011.07.008
4	Benci V , Fortunato D . An eigenvalue problem for the Schrödinger-Maxwell equations. Topological Methods Nonlinear Analysis, 1998, 11: 283- 293 doi: 10.12775/TMNA.1998.019
5	Huang W N , Tang X H . Ground-state solutions for asymptotically cubic Schrödinger-Maxwell equations. Mediterranean Journal of Mathematics, 2016, 13: 3469- 3481 doi: 10.1007/s00009-016-0697-5
6	Tang X H . Non-Nehari manifold method for asymptotically linear Schrödinger equation. Journal of the Australian Mathematical Society, 2015, 98: 104- 116 doi: 10.1017/S144678871400041X
7	Tang X H . Non-Nehari manifold method for superlinear Schrödinger equation. Taiwanese Jounal of Mathematics, 2014, 18: 1957- 1979 doi: 10.11650/tjm.18.2014.3541
8	Kikuchi H . On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Analysis: Theory, Methods and Applications, 2007, 67: 1445- 1456 doi: 10.1016/j.na.2006.07.029
9	Bartsch T, Wang Z Q, Willem M. The Dirichlet problem for superlinear elliptic equations//Mawhin J, et al. Stationary Partial Differential Equations. Amsterdam: Elsevier, 2005, 2: 1-55
10	Azzollini A , D'Avenia P . On a system involving a critically growing nonlinearity. Journal of Mathematical Analysis and Applications, 2012, 387: 433- 438 doi: 10.1016/j.jmaa.2011.09.012
11	Zhao L G , Zhao F K . On the existence of solutions for the Schrödinger-oisson equations. Journal of Mathematical Analysis and Applications, 2008, 346: 155- 169 doi: 10.1016/j.jmaa.2008.04.053
12	Ruiz D . The Schrödinger-Possion equation under the effect of a nonlinear local term. Journal of functional analysis, 2006, 237: 655- 674 doi: 10.1016/j.jfa.2006.04.005
13	Yosida K . Functional Analysis. Berlin, Heidelberg: Springer, 1995
14	Cerami G , Vaira G . Positive solutions for some non-autonomous Schrödinger-Poisson systems. Journal of Differential Equations, 2010, 248: 521- 543 doi: 10.1016/j.jde.2009.06.017
15	Berestycki H , Lions P L . Nonlinear scalar field equations. I. existence of a ground state. Archive for Rational Mechanics and Analysis, 1983, 82: 313- 345 doi: 10.1007/BF00250555
16	Willem M . Minimax Theorems. New York: Springer, 1997
17	Adams R A , Fournier J F . Sobolev Spaces. New York: Academic Press, 2003

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