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Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case

Yu Su, Hichem Hajaiej

Abstract

Our first main contribution consists in establishing an explicit formula of the critical mass via the best constant of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal Laplacian. We also prove the existence of an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. We then show that the optimizer (after some suitable scaling) is an energy ground state solution (with critical mass ). This is a key ingredient to determine sufficient and necessary conditions of existence and non-existence of energy ground state solutions in the in-between critical exponents case. Finally, we show that the energy ground state solution uc0 is an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator.

Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case

Abstract

Our first main contribution consists in establishing an explicit formula of the critical mass via the best constant of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal Laplacian. We also prove the existence of an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. We then show that the optimizer (after some suitable scaling) is an energy ground state solution (with critical mass ). This is a key ingredient to determine sufficient and necessary conditions of existence and non-existence of energy ground state solutions in the in-between critical exponents case. Finally, we show that the energy ground state solution uc0 is an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator.
Paper Structure (9 sections, 20 theorems, 162 equations)

This paper contains 9 sections, 20 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Sobolev Spaces
  4. Proof of Theorem \ref{['Theorem1.1']}
  5. Proof of Theorem \ref{['Theorem1.2']}
  6. Proof of Theorem \ref{['Theorem1.3']}
  7. Proof of Theorem \ref{['Theorem1.4']}
  8. Proof of Theorem \ref{['Theorem1.5']}
  9. Proof of Theorem \ref{['Theorem1.6']}

Key Result

Theorem 1.1

Let $N\geqslant3$, $s\in(0,1)$ and $p\in(2+\frac{4s}{N},2+\frac{4}{N})$. Then there exists $c_{0}>0$ such that equation MLN has an energy ground state solution if and only if $c\geqslant c_{0}$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: Continuous embedded
  • proof
  • Lemma 3.1
  • ...and 30 more