Blow-Up Profile of Pseudo-relativistic Hartree Equations with Singular Potentials

We consider the minimizers for the nonlinear Schrodinger functional E-alpha(u) = integral(R3) (u root-Delta + m(2)u + V(x)u(2))dx - a/2 integral(R3) (vertical bar x vertical bar(-1) * u(2))u(2)dx with the mass constraint parallel to u parallel to(L2(R3)) = 1, for m is an element of R, a > 0 and V(x) is a nonpositive singular potential. Our main results are the existence and blow-up behavior of minimizers when a tends to a critical value a* = parallel to Q parallel to(2)(L2(R3)), where Q is a positive ground state of the following problem root-Delta u + u - (vertical bar x vertical bar(-1) * u(2))u = 0 in R-3, u is an element of H-1/2 (R-3).