一類帶臨界指數(shù)的Schrdinger-Poisson系統(tǒng)的正解的存在性

廖家鋒　陳清方

摘 要：Schrdinger-Poisson系統(tǒng)是由Schrdinger方程和Poisson方程耦合在一起所得，它具有很強(qiáng)的物理背景，主要用來(lái)研究微觀物質(zhì)在電場(chǎng)中的運(yùn)動(dòng)規(guī)律。本文考慮有界區(qū)域上一類帶臨界指數(shù)的Schrdinger-Poisson系統(tǒng)，利用山路定理和Ekeland變分原理，獲得了該系統(tǒng)至少存在2個(gè)正解。

關(guān)鍵詞：Schrdinger-Poisson系統(tǒng)；臨界指數(shù)；山路定理；Ekeland變分原理；正解

中圖分類號(hào)：O177.91 文獻(xiàn)標(biāo)志碼：A 文章編號(hào)：1673-5072（2023）04-0358-07

參考文獻(xiàn)：

［1］ LEI C Y，SUO H M.Positive solutions for a Schrdinger-Poisson system involving concave-convex nonlinearities［J］.Computers & Mathematics with Applications，2017，74（6）：1516-1524.

［2］ HE P F，SUO H M.Positive solutions for Schrdinger-Poisson system with sign changing potential and critical growth［J］.Journal of Function Spaces，2020（86）：1-7.

［3］ ZHENG W X，GAN W B，LIU S B.Existence of positive ground state solutions of Schrdinger-Poisson system involving negative nonlocal term and critical exponent on bounded domain［J］.Boundary Value Problems，2019（1）：1-10.

［4］ LEI C Y.On a Schrdinger-Poisson system with singular term and critical growth［J］.Houston Journal of Mathematics，2019，45（3）：873-892.

［5］ CAI Z P，LEI C Y，CHU C M.On a Schrdinger-Poisson system with singularity and critical nonlinearities［J］.Boundary Value Problems，2020（1）：1-15.

［6］ LEI C Y，LIAO J F.Multiple positive solutions for Schrdinger-Poisson system involving singularity and critical exponent［J］.Mathematical Methods in the Applied Sciences，2019，42（7）：2417-2430.

［7］ ALMUAALEMI B，CHEN H B，KHOUTIR S.Existence of nontrivial solutions for Schrdinger-Poisson system with critical exponent on bounded domains［J］.Bulletin of the Malaysian Mathematical Sciences Society，2019，42（4）：1675-1686.

［8］ 廖家鋒，朱麗君.一類帶有臨界指數(shù)的Schrdinger-Poisson系統(tǒng)非平凡解的存在性［J］.西華師范大學(xué)學(xué)報(bào)（自然科學(xué)版），2022，43（2）：149-155.

［9］ AMBROSETTI A，RUIZ D.Multiple bound states for the Schrdinger-Poisson problem［J］.Communications in Contemporary Mathematics，2018，10（3）：391-404.

［10］SUN J T，WU T F，F(xiàn)ENG Z S.Multiplicity of positive solutions for a nonlinear Schrdinger-Poisson system.［J］.Journal of Differential Equations，2016，260（1）：586-627.

［11］BRZIS H，LIEB E.A relation between pointwise convergence of functions and convergence of functionals［J］.Proceedings of the American Mathematical Society，1983，88：486-490.

［12］BRZIS H，NIRENBERG L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents［J］.Communications on Pure & Applied Mathematics，1983，36（4）：437-477.

［13］MAWHIN J，WILLEM M.Critical Point Theory and Hamiltonian Systems［M］.New York/Berlin：Springer-Verlag，1989.

Existence of Positive Solutionsfor a Class of Schrdinger-Poisson System with Critical Exponent

LIAO Jia-fengab，CHEN Qing-fanga

（a.School of Mathematics & Information，b.College of Mathematics Education，China West Normal University，Nanchong Sichuan 637009，China）

Abstract：Schrdinger-Poisson system is obtained by coupling Schrdinger equation and Poisson equation.With a strong physical background，it is mainly used to study the motion law of micro matter in electric field.In this paper，a class of Schrdinger-Poisson system with critical exponents on bounded domains is considered and two positive solutions are obtained by Mountain Pass Theorem and Ekeland variational principle.

Keywords：Schrdinger-Poisson system;critical exponent;Mountain Pass Theorem;Ekeland variational principle;positive solutions

收稿日期：2021-08-14 基金項(xiàng)目：西華師范大學(xué)創(chuàng)新團(tuán)隊(duì)科研項(xiàng)目（CXTD2018-8）；西華師范大學(xué)基本科研項(xiàng)目（18B015）

作者簡(jiǎn)介：廖家鋒（1983—），男，博士，教授，主要從事非線性泛函分析方面的研究。

通信作者：廖家鋒，E-mail：liaojiafeng@163.com

引文格式：廖家鋒，陳清方.一類帶臨界指數(shù)的Schrdinger-Poisson系統(tǒng)的正解的存在性［J］.西華師范大學(xué)學(xué)報(bào)（自然科學(xué)版），2023，44（4）：358-364.［LIAO J F，CHEN Q F.Existence of positive solutions for a class of Schrdinger-Poisson system with critical exponent［J］.Journal of China West Normal University （Natural Sciences），2023，44（4）：358-364.］