一类与Klein-Gordon-Maxwell问题有关的方程组的基态解的存在性  

The Existence of Ground State Solutions for a Class of Equations Related to Klein-Gordon-Maxwell Systems

作　　者：李易娴 张正杰[1] Li Yixian;Zhang Zhengjie(School of Mathematics and Statistics Central China Normal University,Wuhan 430079)

机构地区：[1]华中师范大学数学与统计学院,武汉430079

出　　处：《数学物理学报（A辑）》2023年第3期680-690,共11页Acta Mathematica Scientia

基　　金：国家自然科学基金(11771166)。

摘　　要：该文利用临界点理论、变分法以及集中紧性原理等理论方法,研究如下一类非线性方程组的基态解的存在性.{−Δu+(m+2ωϕ)u=A(x)|u|^(p−2)u,−Δϕ+λϕ=ωu^(2),lim|_(x|→∞)u(x)=0,lim_(|x|→∞)ϕ(x)=0.其中u∈H^(1)(R^(3)),ϕ∈H^(1)(R^(3)),λ>0,m与ω均为正常数.如果A(x)是正常数,当$4时,上述问题存在基态解(u,ϕ);如果A(x)是非常值函数,当$4时,在适当的情况下上述问题存在基态解(u,ϕ).In this paper,we will study the existence of ground state solutions for a class of nonlinear equations by using the theory of compactness of concentration,variational method and critical point theory..{−Δu+(m+2ωϕ)u=A(x)|u|^(p−2)u,−Δϕ+λϕ=ωu^(2),lim|_(x|→∞)u(x)=0,lim_(|x|→∞)ϕ(x)=0..where u∈H_(1)(R^(3)),ϕ∈H^(1)(R^(3)),λ>0,m andωare positive constants.Then we study the problem assuming the follwwing two cases on A(x).If A(x)is a positive constant function,we prove that the ground state solution(u,ϕ)exists for any p∈(4,6);if A(x)is not a constant function,we prove that the ground state solution(u,ϕ)exists for any p∈(4,6)under the right conditions.

关 键 词：Klein-Gordon-Maxwell 方程 集中紧性原理 变分方法 临界点理论 基态解 

分 类 号：O175.23[理学—数学]