1. Minimizers of mass critical Hartree energy functionals in bounded domains.
- Author
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Guo, Yujin, Luo, Yong, and Zhang, Qi
- Subjects
- *
HARTREE-Fock approximation , *ENERGY function , *MATHEMATICAL bounds , *POTENTIAL theory (Mathematics) , *BOUNDARY value problems - Abstract
We consider L 2 -constraint minimizers of the mass critical Hartree energy functional with a trapping potential V ( x ) in a bounded domain Ω of R 4 . We prove that minimizers exist if and only if the parameter a > 0 satisfies a < a ⁎ = ‖ Q ‖ 2 2 , where Q > 0 is the unique positive solution of − Δ u + u − ( ∫ R 4 u 2 ( y ) | x − y | 2 d y ) u = 0 in R 4 . 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 x 0 of Ω or near the boundary of Ω, depending on whether V ( x ) attains its flattest global minimum at an inner point x 0 of Ω or not. As a byproduct, we also establish two Gagliardo–Nirenberg type inequalities which are of independent interest. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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