Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 18, pp. 1-11.

Title: Existence of high energy solutions for superlinear coupled Klein-Gordons and Born-Infeld equations

Authors: Lixia Wang (Tianjin Chengjian Univ., Tianjin, China)
         Pingping Zhao (Tianjin Chengjian Univ., Tianjin, China)
         Dong Zhang (Tianjin Chengjian Univ., Tianjin, China)

Abstract: 
In this article, we study the system of Klein-Gordon and Born-Infeld equations
$$\displaylines{
-\Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u),  \quad x\in R^3,\cr
\Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad  x\in R^3,
}$$
where $\Delta_4\phi=\hbox{div}(|\nabla\phi|^2\nabla\phi)$,
$\omega$ is a positive constant. Assuming that the primitive of
$f(x,u)$ is of 2-superlinear growth in $u$ at infinity,
we prove the existence of multiple solutions using the fountain theorem.
Here  the potential $V$ are allowed to be a sign-changing function.

Submitted October 30, 2023. Published February 16, 2024.
Math Subject Classifications: 35B33, 35J65, 35Q55.
Key Words: Klein-Gordon equation; Born-Infeld theory; superlinear; fountain theorem.