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

In this paper, we consider the following minimizing problem with two constraints: inf {E(u) | u = (u(1) , u(2)) , ||u(1)||L-2(2) = alpha(1) , ||u(2)||L-2(2) = alpha(2)), where alpha(1) , alpha(2) > 0 and E(u) is defined by E(u) := integral(N)(R)(1/2 Sigma(2)(i=1)|del u(i)|(2) +V-i(x)|u(i)|(2)) - Sigma(2)(i=1) mu(i)/2p(i) + 2 |u(i)|(2pi+2)-beta/p(3)+1 |u(1)|(p3+1)|u(2)|(p3+1)}dx. Here N >= 1, mu(1) , mu(2) , beta > 0 and V-i(x) (i = 1 ,2) are given functions. For V-i(x), we consider two cases: (i) both of V-1 and V-2 are bounded, (ii) one of V-1 and V-2 is bounded. Under some assumptions on Vi and pj , we discuss the compactness of any minimizing sequence.