Limit Behavior of Mass Critical Hartree Minimization Problems With Steep Potential Wells

Authors

Yujin Guo, Chinese Academy of SciencesFollow
Yong Luo, Chinese Academy of SciencesFollow
Zhi-Qiang Wang, Utah State UniversityFollow

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∈H¹(ℝ^d),‖u‖²₂=N} E_λ(u), where d ≥ 3, λ > 0, and the Hartree energy functional E_λ(u) is defined by E_λ(u)≔∫_R^d|∇u(x)|²dx + λ∫_R^d g(x)u²(x)dx − 1/2∫_R^d∫_R^d {u²(x)u²(y)/|x−y|²} dxdy. Here the steep potential g(x) satisfies 0 = g(0) = inf_ℝ^dg(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