In this paper, we consider a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field, namely, the following Klein-Gordon-Maxwell system$\{\begin{array}{c}-\mathrm{\Delta }u+(m^2-{\omega }^2)u-(2\omega +\varphi )\varphi u=f(u),\mathrm{in}{\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }\varphi =-(\omega +\varphi )u^2,\mathrm{in}{\mathbb{R}}^3,\hfill \end{array}$ where $0<\omega <m$ are constants and the nonlinearity f satisfies the superlinear condition. Here m and ω denote the mass and the phase respectively, while u and ϕ are unknowns. By using variational methods and some technique related to Pohozǎev identity, we construct bounded Palais-Smale sequences to obtain the existence of infinitely many high energy radial solutions. Moreover, we show that the ground state solutions of the Klein-Gordon-Maxwell system tend to the ground state solutions of the classical Schrödinger equation as $\omega \to 0$.

在本文中，我们考虑描述与电磁场相互作用的非线性 Klein-Gordon 方程的孤立波的模型，即以下 Klein-Gordon-Maxwell 系统$\{\begin{array}{c}-\mathrm{\Delta }你+({米}^2-{\omega }^2)你-(2\omega +\phi )\phi 你=F(你),\mathrm{在}{\mathbb{电阻}}^3,\hfill \\ -\mathrm{\Delta }\phi =-(\omega +\phi ){你}^2,\mathrm{在}{\mathbb{电阻}}^3,\hfill \end{array}$ 在哪里 $0<\omega <米$是常数，非线性f满足超线性条件。这里m和ω分别表示质量和相位，而u和ϕ是未知数。通过使用变分方法和与 Pohozǎev 恒等式相关的一些技术，我们构造了有界 Palais-Smale 序列以获得无限多个高能径向解的存在性。此外，我们表明 Klein-Gordon-Maxwell 系统的基态解倾向于经典薛定谔方程的基态解为$\omega \to 0$.