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In this paper, we consider the following Klein–Gordon–Maxwell equations−  Δ  u  +  V  (  x  )  u  −  (  2  ω  +  ϕ  )  ϕ  u  =  f  (  x  ,  u  )  +  h  (  x  )inR  3   ,−  Δ  ϕ  +  ϕ   u  2   =  −  ω   u  2inR  3   ,where    ω  >  0    is a constant,  u ,    ϕ  :   R  3   →  R   ,    V  :   R  3   →  R    is a potential function. By assuming the coercive condition on  V  and some new superlinear conditions on  f , we obtain two nontrivial solutions when  h  is nonzero and infinitely many solutions when  f  is odd in  u  and    h  ≡  0    for above equations.

Multiple solutions for superlinear Klein–Gordon–Maxwell equations †