On the concentration of standing waves for NLS equation with point-dipole potential

We study the following minimization problem d(p)(M-p):=inf{E-p(u):parallel to u parallel to L2=M-p} where the Gross-Pitaevskii energy functional E-p(u)=integral(RN)divided by del u divided by(2)-c divided by u divided by(2)/divided by x divided by(2)+V(x)divided by u divided by(2)dx-2/p+2 integral(RN)divided by u divided by(p+2)dx. When p = p & lowast; := 4/N, the precise concentration behavior of minimizers is analyzed as M-p & lowast;NE arrow parallel to Q(p)& lowast;parallel to(L2), where Q(p)* is the unique radially positive solution of -Delta phi-c phi/divided by x divided by(2)-divided by phi divided by(p & lowast;+1)phi=0. When 0 < p < p*, we prove that all minimizers must blow up if lim(p -> p & lowast;)M(p)>= & Vert;Q(p)& lowast;& Vert;(L2). On this argument, the detailed concentration behavior of minimizers is established as p NE arrow p*.