Existence and mass concentration of pseudo-relativistic Hartree equation

In this paper, we investigate the constrained minimization problem e(a): = inf({u is an element of H,parallel to u parallel to 22=1})E(a)(u), where the energy functional E-a(u) = integral(3)(R)(u root-Delta+m(2)u + vu(2)) dx - a/s integral(3)(R) (vertical bar x vertical bar(-1) * u(2))u(2) dx with m is an element of R, a > 0, is defined on a Sobolev space H. We show that there exists a threshold a* > 0 so that e(a) is achieved if 0 < a < a* and has no minimizers if a >= a*. We also investigate the asymptotic behavior of non-negative minimizers of e(a) as a -> a*. Published by AIP Publishing.