Minimizers of mass critical Hartree energy functionals in bounded domains

We consider L2-constraint minimizers of the mass critical Hartree energy functional with a trapping potential V(x) in a bounded domain Ω of R4. We prove that minimizers exist if and only if the parameter a>0 satisfies a0 is the unique positive solution of −Δu+u−(∫R4u2(y)|x−y|2dy)u=0 in R4. By investigating new analytic methods, the refined limit behavior of minimizers as a↗a⁎ is analyzed for both cases where all the mass concentrates either at an inner point x0 of Ω or near the boundary of Ω, depending on whether V(x) attains its flattest global minimum at an inner point x0 of Ω or not. As a byproduct, we also establish two Gagliardo-Nirenberg type inequalities which are of independent interest.