In this paper, we concern the Klein-Gordon-Maxwell system with steep potential well $$\left\{{\matrix{{- {\rm{\Delta u +}}\left({\lambda a\left(x \right) + 1} \right)u - \left({2\omega + \phi} \right)\phi u = f\left({x,u} \right),} \hfill & {{\rm{in}}\,{\mathbb{R}^3},} \hfill \cr {- {\rm{\Delta}}\phi {\rm{=}} - \left({\omega + \phi} \right){u^2},} \hfill & {{\rm{in}}\,{\mathbb{R}^3}.} \hfill \cr}} \right.$$ { − Δu+ ( λ a ( x ) + 1 ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , in ℝ 3 , − Δ ϕ = − ( ω + ϕ ) u 2 , in ℝ 3 . Without global and local compactness, we can tell the difference of multiple solutions from their norms in L P (ℝ 3 ).

中文翻译：

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陡势井 Klein-Gordon-Maxwell 系统的多重解

在本文中，我们关注具有陡峭势阱的 Klein-Gordon-Maxwell 系统 $$\left\{{\matrix{{- {\rm{\Delta u +}}\left({\lambda a\left(x \right) + 1} \right)u - \left({2\omega + \phi} \right)\phi u = f\left({x,u} \right),} \hfill & {{\rm {in}}\,{\mathbb{R}^3},} \hfill \cr {- {\rm{\Delta}}\phi {\rm{=}} - \left({\omega + \phi } \right){u^2},} \hfill & {{\rm{in}}\,{\mathbb{R}^3}.} \hfill \cr}} \right.$$ { − Δu+ ( λ a ( x ) + 1 ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) ，在ℝ 3 ，− Δ ϕ = − (ω + ϕ ) u 2 ，在ℝ 3 。如果没有全局和局部紧凑性，我们可以从它们在 LP 中的范数（ℝ 3 ）中分辨出多个解的差异。