The existence of multiple solutions for the Klein-Gordon equation with concave and convex nonlinearities coupled with Born-Infeld theory on R3

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 {−Δu+a(x)u−(2ω+ϕ)ϕu=λk(x)|u|q−2u+g(x)|u|p−2u,x∈R3,Δϕ+βΔ4ϕ=4π(ω+ϕ)u2,x∈R3, where 1