一类 Klein-Gordon-Maxwell 系统解的存在性和多重性
Existence and Multiplicity of Solutions to a Class of Klein-Gordon-Maxwell Systems
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收稿日期: 2024-08-13 修回日期: 2025-01-13
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Received: 2024-08-13 Revised: 2025-01-13
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研究如下 Klein-Gordon-Maxwell 系统
关键词:
This article concerns the following Klein-Gordon-Maxwell system
Keywords:
本文引用格式
段誉, 孙歆.
Duan Yu, Sun Xin.
1 引言
研究如下 Klein-Gordon-Maxwell 系统
其中
其中
(V)
(K
(K
(F
(F
本文的主要结果如下
定理 1.1 (i) 假设 (V), (K
(ii) 假设 (V), (K
2 预备知识
令
其范数为
定义
其范数为
显然在条件
引理 2.1[8]
对任何
(i) 在集合
(ii)
设
其中
其中
引理 2.2[27] 假设 (F
(i) 当
(ii)
(iii)
现考虑修正后的如下 Klein-Gordon-Maxwell 系统
按照常规做法 (见文献 [8] 等) 知, 系统 (2.1) 对应的泛函为
由引理 2.1 及引理 2.2 知,
由命题 3.5[1] 知,
引理 2.3[29]
假设
引理 2.4[30]
假设
其中
记
3 定理 1.1 的证明
为了证明定理 1.1, 下面给出两个引理.
引理 3.1 假设 (V), (K
证 由引理 2.1 及引理 2.2 知,
结合
由
对任意给定的
及
其中
由引理 2.2 及
由 (3.2)-(3.5) 式及引理 2.1(i) 知
即 (3.1) 式成立.
其次证明:
在空间
由于在
所以要证:
易知
由于对任意的
同理可证
故由 (3.7)-(3.11) 式知: 在空间
引理 3.2 假设 (V), (K
证 因为
对任意的
易知
令
则
由 (3.13)-(3.15) 式知
从而由 (3.12) 式及 (3.16)-(3.17) 式知
故
进而结合引理 2.2, (K
令
即
下面就 (3.20) 式利用 Moser 迭代说明
在 (3.20) 式中取
在 (3.20) 式中取
由 Moser 迭代知
由 (3.22)-(3.23) 式知,
故
由(3.21) 式及 (3.23) 式知
由 (3.27) 式知
根据达朗贝尔判别法易知正项级数
在 (3.26) 式令
定理 1.1 的证明 (i) 显然
结合
下证:
即
从而结合引理 3.2 知
这意味着: 存在常数
(ii) 由引理 2.2 知,
事实上, 对任意的
由引理 2.1, 引理 2.2 及有限维空间范数等价性知, 对任意的
由
参考文献
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Elliptic systems with a partially sublinear local term
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渐近线性 Klein-Gordon-Maxwell 系统正解的存在性
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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
Positive ground state solutions for quasicritical the fractional Klein-Gordon-Maxwell system with potential vanishing at infinity
DOI:10.1080/17476933.2018.1434625
This paper deals with the fractional Klein-Gordon-Maxwell system when the nonlinearity has a quasicritical growth at infinity, where V(x) is bounded or involving zero mass potential, that is, when V(x) -> 0, as vertical bar x vertical bar -> infinity. The interaction of the behaviour of the potential and nonlinearity recover the lack of the compactness of Sobolev embedding in whole space. The positive ground state solution is obtained by proving that the solution satisfies the Mountain Pass level.
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带有凹凸非线性项的 Klein-Gordon-Maxwell 系统解的存在性和多重性
一类 Klein-Gordon-Maxwell 方程无穷多解的存在性
Existence of infinitely many solutions for a Klein-Gordon-Maxwell system
带有次线性项和超线性项的 Klein-Gordon-Maxwell 系统多重解的存在性
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