广西师范大学学报（自然科学版） ›› 2023, Vol. 41 ›› Issue (6): 113-121.doi: 10.16088/j.issn.1001-6600.2023020101

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一类带双临界指数的凹凸非线性分数阶Schrödinger-Poisson系统的两个正解

蒋维, 李雨涵, 李红英*   

  1. 西华师范大学 数学与信息学院, 四川 南充 637009
  • 收稿日期:2023-02-01 修回日期:2023-07-17 发布日期:2023-12-04
  • 通讯作者: 李红英(1984—), 女, 四川南充人, 西华师范大学副教授。 E-mail: lihongyingnch@163.com
  • 基金资助:
    四川省自然科学基金(2022NSFSC1816)

Two Positive Solutions for a Class of Concave-convex Fractional Schrödinger-Poisson System with Doubly Critical Exponents

JIANG Wei, LI Yuhan, LI Hongying*   

  1. School of Mathematics and Information, China West Normal University, Nanchong Sichuan 637009, China
  • Received:2023-02-01 Revised:2023-07-17 Published:2023-12-04

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摘要： 本文研究如下带双临界指数的凹凸非线性分数阶Schrödinger-Poisson系统
$\left\{\begin{array}{lr}(-\Delta)^s \boldsymbol{u}-\phi|\boldsymbol{u}|^{2_s^*-3} \boldsymbol{u}=|\boldsymbol{u}|^{2_s^*-2} \boldsymbol{u}+\lambda h(\boldsymbol{x})|\boldsymbol{u}|^{q-2} \boldsymbol{u}, & \boldsymbol{x} \in \mathbf{R}^3, \\ (-\Delta)^s \phi=|\boldsymbol{u}|^{2_s^*-1}, & \boldsymbol{x} \in \mathbf{R}^3,\end{array}\right.$
式中:10是实参数; h为满足一定条件的函数。利用变分法和山路定理,本文证明存在λ*>0,使得当λ∈(0,λ*)时,该系统在Ds,2(R3)中存在1个具有负能量的局部极小正解和1个具有正能量的山路解。

关键词: 分数阶Schrödinger-Poisson系统, 正解, 山路定理, 双临界指数

Abstract: In this paper, the following concave-convex fractional Schrödinger-Poisson system with doubly critical exponents is investigated
$\left\{\begin{array}{lr}(-\Delta)^s \boldsymbol{u}-\boldsymbol{\phi}|\boldsymbol{u}|^{2_s^*-3} \boldsymbol{u}=|\boldsymbol{u}|^{2_s^*-2} \boldsymbol{u}+\lambda h(\boldsymbol{x})|\boldsymbol{u}|^{q-2} \boldsymbol{u}, & \boldsymbol{x} \in \mathbf{R}^3, \\ (-\Delta)^s \boldsymbol{\phi}=|\boldsymbol{u}|^{2_s^*-1}, & \boldsymbol{x} \in \mathbf{R}^3,\end{array}\right.$
where 10 is a real parameter and h satisfies some certain conditions. It is showed that there exists λ*>0 such that the system has a positive local minima solution with negative energy and a positive mountain-pass solution with positive energy for any λ∈(0,λ*) in Ds,2(R3) by applying the Mountain Pass Theorem and variational method.

Key words: fractional Schrödinger-Poisson system, positive solutions, mountain pass theorem, doubly critical exponents

中图分类号:  O177.91

[1] LANDKOF N. Foundations of modern potential theory[M]. Berlin: Springer-Verlag, 1972.
[2] CAFFARELLI L, VALDINOCI E. Uniform estimates and limiting arguments for nonlocal minimal surfaces[J]. Calculus of Variations and Partial Differential Equations, 2011, 41(1): 203-240. DOI: 10.1007/s00526-010-0359-6.
[3] LASKIN N. Fractional quantum mechanics and Lévy path integrals[J]. Physics Letters A, 2000, 268(4/5/6): 298-305. DOI: 10.1016/S0375-9601(00)00201-2.
[4] LIU H D. Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent[J]. Nonlinear Analysis: Real World Applications, 2016, 32: 198-212. DOI: 10.1016/j.nonrwa.2016.04.007.
[5] LI N, HE X M. Existence and multiplicity results for some Schrödinger-Poisson system with critical growth[J]. Journal of Mathematical Analysis and Applications, 2020, 488(2): 124071. DOI: 10.1016/j.jmaa.2020.124071.
[6] DOU X L, HE X M. Ground states for critical fractional Schrödinger-Poisson systems with vanishing potentials[J]. Mathematical Methods in the Applied Sciences, 2022:45(16): 9089-9110. DOI: 10.1002/mma.8294.
[7] 张炫, 冯晓晶. 带临界项的分数阶薛定谔-泊松系统的非平凡解[J]. 纺织高校基础科学学报, 2021, 34(4): 95-101. DOI: 10.13338/j.issn.1006-8341.2021.04.014.
[8] 郭凯利, 冯晓晶. 具有临界项的分数阶薛定谔-泊松系统的解[J]. 山东大学学报(理学版), 2021, 56(6): 56-63. DOI: 10.6040/j.issn.1671-9352.0.2020.713.
[9] 张鑫瑞, 贺小明, 屈思琪. 具有临界频率的分数阶薛定谔-泊松系统的半经典基态解的多重性[J]. 中央民族大学学报(自然科学版), 2021, 30(1): 21-27. DOI: 10.3969/j.issn.1005-8036.2021.01.005.
[10] 张鑫瑞. 分数阶薛定谔-泊松系统解的存在性, 集中性与多解性的研究[D]. 北京: 中央民族大学, 2021.
[11] 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. DOI: 10.3969/j.issn.1003-3998.2020.06.013.
[12] HE X M. Positive solutions for fractional Schrödinger-Poisson systems with doubly critical exponents[J]. Applied Mathematics Letters, 2021, 120: 107190. DOI: 10.1016/j.aml.2021.107190.
[13] FENG X J. Nontrivial solution for Schrödinger-Poisson equations involving the f ractional Laplacian with critical exponent[J]. Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A. Matemáticas, 2021, 115(1): 10. DOI: 10.1007/s13398-020-00953-w.
[14] FENG X J, YANG X. Existence of ground state solutions for fractional Schrödinger-Poisson systems with doubly critical growth[J]. Mediterranean Journal of Mathematics, 2021, 18(2): 41. DOI: 10.1007/S00009-020-01660-X.
[15] FENG X J. Ground state solutions for Schrödinger-Poisson systems involving the fractional Laplacian with critical exponent[J]. Journal of Mathematical Physics, 2019, 60(5): 051511. DOI: 10.1063/1.5088710.
[16] JIANG W, LIAO J F. Multiple positive solutions for fractional Schrödinger-Poisson system with doubly critical exponents[J]. Qualitative Theory of Dynamical Systems, 2023, 22(1): 25. DOI: 10.1007/S12346-022-00726-3.
[17] WILLEM M. Minimax Theorems[M]. Boston: Birkhauser, 1996. DOI: 10.1007/978-1-4612-4146-1.
[18] BRÉZIS H, LIEB E. A relation between pointwise convergence of functions and convergence of functionals[J]. Proceedings of the American Mathematical Society, 1983, 88(3): 486-490. DOI: 10.1090/S0002-9939-1983-0699419-3.
[19] SERVADEI R, VALDINOCI E. The Brézis-Nirenberg result for the fractional Laplacian[J]. Transactions of the American Mathematical Society, 2015, 367(1): 67-102. DOI: 10.1090/S0002-9947-2014-05884-4.
[20] AMBROSETTI A, RABINOWITZ P H. Dual variational methods in critical point theory and applications[J]. Journal of Functional Analysis, 1973, 14(4): 349-381. DOI: 10.1016/0022-1236(73)90051-7.
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