Existence of two solutions for nonhomoge- neous Klein-Gordon equation with sign-changing potential coupled with Born- Infeld theory

Abstract

In this paper, we study the following nonhomogeneous Klein-Gordon equation with Born-Infeld theory −∆u + λV (x)u − K(x)(2ω + φ)φu = f (x, u) + h(x), x ∈ R 3 , ∆φ + β∆4φ = 4πK(x)(ω + φ)u 2 , x ∈ R 3 , where ω > 0 is a constant, λ > 0 is a parameter and 4φ = div(||φ| 2 φ). Under some suitable assumptions on V, K, f and h, the existence of multiple solutions is proved by using the Linking theorem and the Ekeland’s variational principle in critical point theory. Especially, the potential V is allowed to be sign-changing. Mathematics Subject Classification (2010). Primary 35B33; Secondary 35J65.