Multiple solutions for the nonhomogeneous Klein–Gordon equation coupled with Born–Infeld theory on R3

In this paper we prove the existence of multiple solutions for the following nonhomogeneous Klein–Gordon equation coupled with Born–Infeld theory, { − Δ u + [ m 2 − ( ω + ϕ ) 2 ] u = | u | q − 2 u + h ( x ) , in R 3 , Δ ϕ + β Δ 4 ϕ = 4 π ( ω + ϕ ) u 2 , in R 3 , when 2 q 6 .