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Mixed-dispersion Schrödinger equations and Gagliardo-Nirenberg inequalities: equivalence between ground states and optimizers

Zhisu Liu, Giulio Romani, Yu Su

Abstract

We study a nonlinear Schrödinger equation with mixed dispersion in the mass competition regime, namely mass-supercritical for the Laplacian and mass-subcritical for the Bilaplacian. In this setting, the existence of a critical value of the mass $c_\varepsilon$, which divides existence and nonexistence of energy ground state solutions, was established in [Bonheure, Castéras, dos Santos, Nascimento, SIAM J. Math. Anal. 50 (2018)]. In this work, we strengthen these results by investigating the relationship between the energy ground states with critical mass, and the optimizers of mixed Gagliardo-Nirenberg-type inequalities. Moreover, we discuss the equivalence between energy and action ground states solutions.

Mixed-dispersion Schrödinger equations and Gagliardo-Nirenberg inequalities: equivalence between ground states and optimizers

Abstract

We study a nonlinear Schrödinger equation with mixed dispersion in the mass competition regime, namely mass-supercritical for the Laplacian and mass-subcritical for the Bilaplacian. In this setting, the existence of a critical value of the mass , which divides existence and nonexistence of energy ground state solutions, was established in [Bonheure, Castéras, dos Santos, Nascimento, SIAM J. Math. Anal. 50 (2018)]. In this work, we strengthen these results by investigating the relationship between the energy ground states with critical mass, and the optimizers of mixed Gagliardo-Nirenberg-type inequalities. Moreover, we discuss the equivalence between energy and action ground states solutions.

Paper Structure

This paper contains 10 sections, 18 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation
  4. Preliminaries
  5. First characterization of the energy GSS with critical mass c_epsilon: Proof of Theorem \ref{['Theorem1.1']}
  6. Second characterization of the energy GSS with critical mass c_epsilon: Proof of Theorem \ref{['Theorem1.2']}
  7. Existence of optimizers for GN_C (Proof of Proposition \ref{['Theorem4.1']})
  8. Any optimizer is an energy GSS with critical mass c_epsilon (Proof of Proposition \ref{['Theorem4.2']})
  9. Any energy GSS with critical mass c_epsilon is an optimizer for GN_C (Proof of Proposition \ref{['Theorem4.4']})
  10. On the relationship between action GSS and energy GSS: Proof of Theorem \ref{['Theorem1.3']}

Key Result

Theorem A

Cazenave-Lions1982CMPWeinstein1983CMP Let $N\geqslant 1$.

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 24 more