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

In this article, we study the system of Klein-Gordon and Born-Infeld equations $$\displaylines{ -\Delta u +V(x)u-(2\omega+\phi)\phi u =f(x,u), \quad x\in \mathbb{R}^3,\cr \Delta \phi+\beta\Delta_4\phi=4\pi(\omega+\phi)u^2, \quad x\in \mathbb{R}^3, }$$ 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. For more information see https://ejde.math.txstate.edu/Volumes/2024/18/abstr.html