Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 74, pp. 1-11.

Title: Two solutions for nonhomogeneous Klein-Gordon equations coupled 
with Born-Infeld type equations
       
Authors: Lixia Wang     (Tianjin Chengjian Univ., Tianjin, China)
         Chunlian Xiong (Tianjin Chengjian Univ., Tianjin, China)
         Pingping Zhao  (Tianjin Chengjian Univ., Tianjin, China)
         
Abstract: 
This article concerns the nonhomogeneous Klein-Gordon equation
coupled with a Born-Infeld type equation,
$$\displaylines{
- \Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u)+h(x),  \quad x\in \mathbb{R}^3,\cr
\Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad  x\in \mathbb{R}^3,
}$$
where $\omega$ is a positive constant.
We obtain the existence of two solutions using the Mountain Pass Theorem,
and the Ekeland's variational principle in critical point theory.

Submitted August 2, 2022. Published November 15, 2022.
Math Subject Classifications: 35B33, 35J65, 35Q55.
Key Words: Klein-Gordon equation; Born-Infeld theory; nonhomogeneous;
           Mountain pass theorem; Ekeland's variational principle.