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On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

Bin Chen, Yinbin Deng, Yujin Guo, Chenyang Wang

Abstract

We consider ground states of a pseudo-relativistic Fermi system in the $L^2$-critical case. We prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<D_{4/3,2}$, where $D_{4/3,2}\in(0, \infty)$ is the optimal constant of a dual fractional Lieb--Thirring inequality. The limiting behavior of ground states for the system is further analyzed as $a\nearrow D_{4/3,2}$. As a byproduct, the qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are also investigated.

On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

Abstract

We consider ground states of a pseudo-relativistic Fermi system in the -critical case. We prove that the system admits ground states, if and only if the attractive strength satisfies , where is the optimal constant of a dual fractional Lieb--Thirring inequality. The limiting behavior of ground states for the system is further analyzed as . As a byproduct, the qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are also investigated.
Paper Structure (7 sections, 10 theorems, 179 equations)

This paper contains 7 sections, 10 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A Dual Fractional Lieb--Thirring Inequality
  4. Existence and Non-existence of Minimizers
  5. Limiting Behavior of Minimizers as $a\nearrow D_{4/3,N}$
  6. Proof of Theorem \ref{['th22']}
  7. Appendix

Key Result

Theorem 1.1

Let $E_{a}(2)$ and $D_{4/3,2}$ be defined by $(problem)$ and $(LTineq)$, respectively. Then we have the following conclusions:

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3