Document Type

Article

Journal/Book Title/Conference

Journal of Mathematical Physics

Volume

59

Issue

6

Publisher

American Institute of Physics

Publication Date

6-7-2018

First Page

1

Last Page

19

DOI

https://doi.org/10.1063/1.5025730

Abstract

We consider minimizers of the following mass critical Hartree minimization problem: eλ(N) ≔ inf{u∈H1(ℝd),‖u‖22=N} Eλ(u), where d ≥ 3, λ > 0, and the Hartree energy functional Eλ(u) is defined by Eλ(u)≔∫Rd|∇u(x)|2dx + λ∫Rd g(x)u2(x)dx − 1/2∫RdRd {u2(x)u2(y)/|x−y|2} dxdy. Here the steep potential g(x) satisfies 0 = g(0) = infdg(x) ≤ g(x) ≤ 1 and 1 − g(x)∈Ld/2(ℝd). We prove that there exists a constant N* > 0, independent of λg(x), such that if NN*, then eλ(N) does not admit minimizers for any λ > 0; if 0 < N < N*, then there exists a constant λ*(N) > 0 such that eλ(N) admits minimizers for any λ > λ*(N) and eλ(N) does not admit minimizers for 0 < λ < λ*(N). For any given 0 < N < N*, the limit behavior of positive minimizers for eλ(N) is also studied as λ → ∞, where the mass concentrates at the bottom of g(x).

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