Two solutions for nonhomogeneous Klein-Gordon equations coupled with Born-Infeld type equations

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.

Existence of solutions for critical Klein–Gordon equations coupled with Born–Infeld theory in higher dimensions

“…In recent years, the Born-Infeld nonlinear electromagnetism has regained its importance due to its relevance in the theory of superstring and membranes. When f .u/ D juj p 2 u, d'Avenia and Pisani [12] established the following system that has infinitely many radially symmetric solutions under the assumptions jmj > ! and 4 < p < 6:…”