In this paper we consider the following Klein-Gordon equation coupled with the Born-Infeld theory \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + (\lambda A(x) + 1)u - (2\omega + \phi )\phi u = f(x,u)\ & \text { in }{\mathbb {R}}^3, \\ \Delta \phi + \beta \Delta _{4}\phi =4\pi (\omega +\phi )u^2\ & \text { in }{\mathbb {R}}^3,\\ \end{array} \right. \end{aligned}\) where f may be a superlinear term, or it may be an asymptotically linear term. When f satisfies the superlinear conditions, we obtain the existence of a ground state solution. When f satisfies the asymptotic conditions, we prove the existence of positive solutions based on variational methods and some analytical techniques. In addition, we study the properties of decay estimates and asymptotic behavior for the positive solution.
