The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

In this paper, we consider the following minimizing problem with two constraints: inf E(u)|u = (u1,u2),u1L22 = α 1,u2L22 = α 2, where α1,α2 > 0 and E(u) is defined by E(u):= N 1 2i=12(|u i|2 + V i(x)|ui|2) - i=12 μi 2pi + 2|ui|2pi+2 - β p3 + 1|u1|p3+1|u 2|p3+1 dx. Here N ≥ 1, μ1,μ2,β > 0 and Vi(x) (i = 1, 2) are given functions. For Vi(x), we consider two cases: (i) both of V1 and V2 are bounded, (ii) one of V1 and V2 is bounded. Under some assumptions on Vi and pj, we discuss the compactness of any minimizing sequence.