Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth

TL;DR

This paper establishes the existence of nontrivial solutions for a coupled Klein-Gordon and Born-Infeld system with critical growth, using variational methods and iterative techniques, also applicable to Klein-Gordon-Maxwell systems.

Contribution

It introduces a novel approach combining cut-off functions and Moser iteration to prove solutions without growth restrictions or Ambrosetti-Rabinowitz conditions.

Findings

Existence of nontrivial solutions for the system.

Method applicable to Klein-Gordon-Maxwell system.

Handles critical growth without standard conditions.

Abstract

In this paper, we study the following system \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u + V(x)u-(2\omega+\phi)\phi u=\lambda f(u)+|u|^{4}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{eqnarray*} where $f(u)$ without any growth and Ambrosetti-Rabinowitz conditions. We use cut-off function and Moser iteration to obtain the existence of nontrivial solution. Finally, as a by-product of our approaches, we get the same result for Klein-Gordon-Maxwell system.