In this paper,we concern the Klein-Gordon-Maxwell system with steep potential well{-Δu +(λa(x)+ 1)u-(2ω + φ)φu = f(x,u),in R3,-Δφ=-(ω+φ)u2,in R3.Without global and local compactness,we can tell the diference of multiple solutions from their norms in Lp(R3).