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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.

In this paper we study the following nonhomogeneous Klein–Gordon–Maxwell equations { − △ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) + h ( x ) , in R 3 , − △ ϕ + ϕ u 2 = − ω u 2 , in R 3 . Under appropriate assumptions on V , f and h , the existence of multiple solutions is proved via Variational methods.

In this paper we study the following nonhomogeneous Kirchhoff equation − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) + h ( x ) , in R N , where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V , f and h , the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory.