## 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

## Abstract

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

## Recommended Citation

Guo, Yujin; Luo, Yong; and Wang, Zhi-Qiang, "Limit Behavior of Mass Critical Hartree Minimization Problems With Steep Potential Wells" (2018). *Mathematics and Statistics Faculty Publications.* Paper 234.

https://digitalcommons.usu.edu/mathsci_facpub/234