Minimizers of mass critical Hartree energy functionals in bounded domains

摘要

We consider L-2-constraint minimizers of the mass critical Hartree energy functional with a trapping potential V (x) in a bounded domain Omega of R-4. We prove that minimizers exist if and only if the parameter a 0 satisfies a a* = parallel to Q parallel to(2)(2), where Q 0 is the unique positive solution of -Delta u + u - (f(R4) u(2)(y)/vertical bar x-y vertical bar(2) dy)u = 0 in R-4. By investigating new analytic methods, the refined limit behavior of minimizers as a NE arrow a* is analyzed for both cases where all the mass concentrates either at an inner point x(0) of Omega or near the boundary of Omega depending on whether V (x) attains its flattest global minimum at an inner point x(0) of Omega or not. As a byproduct, we also establish two Gagliardo-Nirenberg type inequalities which are of independent interest. (C) 2018 Elsevier Inc. All rights reserved.