Existence and mass concentration of pseudo-relativistic Hartree equation.

In this paper, we investigate the constrained minimization problem e(a):=inf{u∊H ∣∣u∣∣22=1}Ea(u), where the energy functional Ea(u) = ∫ℝ3(u√-Δ + m2 u + Vu2)dx-a/2∫ℝ3(∣x∣-1 *u2)u2dx with m∊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*. [ABSTRACT FROM AUTHOR]