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Attainability and criticality for multipolar Rellich inequality

Yongyang Jin, Shoufeng Shen, Li Tang

Abstract

In this paper we obtain optimal multipolar Rellich inequality for biharmonic Schrodinger operator with positive multi-singular potentials. Moreover, we prove the attainability of the best constant and the criticality of the biharmonic Schrodinger operator.

Attainability and criticality for multipolar Rellich inequality

Abstract

In this paper we obtain optimal multipolar Rellich inequality for biharmonic Schrodinger operator with positive multi-singular potentials. Moreover, we prove the attainability of the best constant and the criticality of the biharmonic Schrodinger operator.

Paper Structure

This paper contains 5 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. $L^2$-case multipolar Rellich inequality with sharp constant
  3. Attainability and Criticality
  4. Attainability for constant $\frac{N^{2}(N-4)^{2}}{n^{4}}$
  5. Criticality for $n=2$

Key Result

Theorem \oldthetheorem

Assume $N\geq 5$, $a_{1},\cdots,a_{n}\in\mathbb{R}^{N}$. Then the following inequality holds for any $u\in C_{0}^{\infty}(\mathbb{R}^{N})$, where with The constant $\frac{N^{2}(N-4)^{2}}{n^{4}}$ is sharp. Furthermore, the constant $\frac{N^{2}(N-4)^{2}}{n^{4}}$ is achieved in the space $D^{2,2}(\mathbb{R}^{N})$ when $n\geq3$ by the minimizers (up to a constant) while it can not be attained in

Theorems & Definitions (16)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • ...and 6 more