Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency

In this paper, we are concerned with the existence and asymptotic behavior of standing wave solutions ψ(x,t)=e−iλEt of nonlinear Schrödinger equations with electromagnetic fields , , with E being a critical frequency in the sense that . We show that if the zero set of W−E has several isolated connected components Ω1,…,Ωk such that the interior of Ωi is not empty and ∂Ωi is smooth, then for λ>0 large there exists, for any non-empty subset J {1,2,…,k}, a standing wave solution which is trapped in a neighborhood of j JΩj.