Klein–Gordon equation

Algorithm

Combinatorics

Physics

Quadratic equation

Mathematical analysis

Nonlinear system

Geometry

Omega

Mathematics

Mathematical physics

State (computer science)

Quantum mechanics

Ground state

Infinity

Discrete and Continuous Dynamical Systems

This paper is concerned with the following Klein-Gordon-Maxwell system \begin{document}$\left\{ \begin{align} &-\vartriangle u+\left[ m_{0}^{2}-{{(\omega +\phi )}^{2}} \right]u = f(u),\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ &\vartriangle \phi = (\omega +\phi ){{u}^{2}},\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ \end{align} \right.$ \end{document} where $0 < ω≤ m_0$ and $f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$. By introducing some new tricks, we prove that the above system has 1) a ground state solution in the case when $0 < ω < m_0$ and $f$ is superlinear at infinity; 2) a nontrivial solution in the zero mass case, i.e. $ω = m_0$ and $f$ is super-quadratic at infinity. These results improve the related ones in the literature.

Improved results for Klein-Gordon-Maxwell systems with general nonlinearity