Constraint minimizers of mass critical Hartree energy functionals: Existence and mass concentration

Abstract

We consider L2-constraint minimizers of mass critical Hartree energy functionals in ℝN with N ≥ 3. We prove that minimizers exist if and only if the parameter a > 0 satisfies a<a∗=Q22, where Q is a positive radially symmetric ground state of Δu−u+∫RNu(y)2x−y2dyu=0 in ℝN. The blow-up behavior of minimizers as a approaches a∗ is also analyzed, for which all the mass concentrates at a global minimum point x0 of the external potential V(x).