带有双临界项的薛定谔-泊松系统非平凡解的存在性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1590-1598.

带有双临界项的薛定谔-泊松系统非平凡解的存在性

冯晓晶()

山西大学数学科学学院太原 030006

收稿日期:2019-11-06 出版日期:2020-12-26 发布日期:2020-12-29
作者简介:冯晓晶, E-mail: fengxj@sxu.edu.cn
基金资助:
国家自然科学基金(11571209);国家自然科学基金(11671239);山西省自然科学基金(201801D211001);山西省自然科学基金(201801D121002);山西省自然科学基金(201801D221012);山西省留学基金(020-005)

Nontrivial Solution for Schrödinger-Poisson type Systems with Double Critical Terms

Xiaojing Feng()

School of Mathematical Sciences, Shanxi University, Taiyuan 030006

Received:2019-11-06 Online:2020-12-26 Published:2020-12-29
Supported by:
the NSFC(11571209);the NSFC(11671239);NSF of Shanxi Province(201801D211001);NSF of Shanxi Province(201801D121002);NSF of Shanxi Province(201801D221012);the Shanxi Scholarship Council(020-005)

摘要/Abstract

摘要：

该文利用集中紧性原理和山路定理, 研究了一类具有非线性临界增长和非局部临界增长的薛定谔-泊松系统非平凡解的存在性.系统中的双临界增长项对有界(PS)序列的收敛性造成一定的困难.此外, 山路的水平也较难估计.该文关键之处在于证明山路的临界水平值低于相应能量泛函的非紧性水平.

关键词: 薛定谔-泊松系统, 临界指数, 变分方法

Abstract:

The existence of a nontrivial solution to the Schrödinger-Poisson type system with both nonlinear critical growth and nonlocal critical growth is obtained by applying the concentration-compactness principle and mountain pass theorem. The double critical growth in the system presents an obstacle when showing the convergence of the bounded (PS) sequences. At the same time, it is difficult to estimate the critical level of the mountain pass. The key ingredient of the paper is to show that the critical level of the mountain pass is below the non-compactness level of the associated energy functional.

Key words: Schrödinger-Poisson type system, Critical exponent, Variational method

中图分类号:

O175.2

冯晓晶. 带有双临界项的薛定谔-泊松系统非平凡解的存在性[J]. 数学物理学报, 2020, 40(6): 1590-1598.

Xiaojing Feng. Nontrivial Solution for Schrödinger-Poisson type Systems with Double Critical Terms[J]. Acta mathematica scientia,Series A, 2020, 40(6): 1590-1598.

参考文献 29

1	Azzollini A , Pomponio A . Ground state solutions for nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345: 90- 108
2	Benci V , Fortunato D . An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11: 283- 293
3	Brezis H , Nirenberg L . Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36: 437- 477
4	Candela A , Salvatore A . Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations. Mediterr J Math, 2006, 3: 483- 493
5	Cerami G , Vaira G . Positive solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2010, 248: 521- 543
6	Chen S , Tang X . Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $ {{\mathbb{R}}^{3}} $. Z Angew Math Phys, 2016, 4: 1- 18
7	Duan S , Wu X . Multiplicity of solutions for a class of Schrödinger-Maxwell systems. Mediterr J Math, 2017, 14: 197
8	Fang L W , Huang W N , Wang M Q . Ground-state solutions for Schrödinger-Maxwell equations in the critical growth (in Chinese). Acta Mathematica Scientia, 2019, 39A (3): 475- 483
9	Feng X . Ground state solution for a class of Schrödinger-Poisson-type systems with partial potential. Z Angew Math Phys, 2020, 71: 37
10	He Y , Li G . Standing waves for a class of Schrödinger-Poisson equations in $ {{\mathbb{R}}^{3}} $ involving critical Sobolev exponents. Ann Acad Sci Fenn Math, 2015, 40 (2): 729- 766
11	He X , Zou W . Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J Math Phys, 2012, 53: 023702
12	Huang L , Rocha E , Chen J . Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity. J Math Anal Appl, 2013, 408: 55- 69
13	Ianni I . Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem. Topol Methods Nonlinear Anal, 2013, 41: 365- 385
14	Kim S , Seok J . On nodal solutions of the nonlinear Schrödinger-Poisson equations. Commun Contemp Math, 2012, 14 (6): 1250041
15	Liu H . Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent. Nonlinear Anal RWA, 2016, 32: 198- 212
16	Li F , Li Y , Shi J . Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent. Comm Contem Math, 2014, 16: 1450036
17	Liu Z L , Wang Z Q , Zhang J J . Infnitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Ann Mat, 2016, 195: 775- 794
18	Liu Z , Guo S . On ground state solutions for the Schrödinger-Poisson equations with critical growth. J Math Anal Appl, 2014, 412: 435- 448
19	Ruiz D . The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237: 655- 674
20	Shuai W , Wang Q F . Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $ {{\mathbb{R}}^{3}} $ . Z Angew Math Phys, 2015, 66: 3267- 3282
21	Sun J . Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations. J Math Anal Appl, 2012, 390: 514- 522
22	Tang X , Chen S . Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Disc Contin Dyn Syst Ser A, 2017, 37 (9): 4973- 5002
23	Wang Z , Zhou H . Positive solution for a nonlinear stationary Schrödinger-Poisson system in $ {{\mathbb{R}}^{3}} $. Discrete Contin Dyn Syst, 2007, 18: 809- 816
24	Wang Z , Zhou H . Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $ {{\mathbb{R}}^{3}} $. Calc Var Partial Differ Equ, 2015, 52: 927- 943
25	Willem M . Minimax Theorems. Boston: Birkhäuser, 1996
26	Zhao L , Zhao F . Positive solutions for Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal, 2009, 70: 2150- 2164
27	Zhang J . On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth. Nonlinear Anal, 2012, 75: 6391- 6401
28	Zhang J . Ground state and multiple solutions for Schrödinger-Poisson equations with critical nonlinearity. J Math Anal Appl, 2016, 440: 466- 482
29	Zhong X J , Tang C L . Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $ {{\mathbb{R}}^{3}} $. Nonlinear Anal RWA, 2018, 39: 166- 184

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