The existence of multiple solutions for the Klein–Gordon equation with concave and convex nonlinearities coupled with Born–Infeld theory onR3

In this paper, we prove the existence of multiple solutions for the following Klein–Gordon equation with concave and convex nonlinearities coupled with Born–Infeld theory { − Δ u + a ( x ) u − ( 2 ω + ϕ ) ϕ u = λ k ( x ) | u | q − 2 u + g ( x ) | u | p − 2 u , x ∈ R 3 , Δ ϕ + β Δ 4 ϕ = 4 π ( ω + ϕ ) u 2 , x ∈ R 3 , where 1 q 2 p 6 . Under appropriate assumptions on a , k , g and λ , the existence of multiple nontrivial solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory.