Existence and concentration of solutions for Schrödinger–Poisson system with steep potential well

This paper is concerned with the nonlinear Schrödinger–Poisson system− Δ u + λV ( x ) u + K ( x ) ϕu = f ( x , u ) ,x ∈R 3 ,− Δ ϕ = K ( x ) u 2 ,x ∈R 3 ,where λ > 0 is a parameter. We require that V ≥0 and has a bounded potential well V −1 (0). Combining this with other suitable assumptions on K and f , the existence of nontrivial solutions is obtained by using variational methods. Moreover, the concentration of solutions is also explored on the set V −1 (0) as λ → ∞ . Copyright © 2015 John Wiley & Sons, Ltd.