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

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