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Nonlinear Schrödinger equations with critical Hardy potential and Choquard nonlinearity

Phuoc-Tai Nguyen, Tuan Dat Tran

Abstract

We study the Cauchy problem for the nonlinear Schrödinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Poho\v zaev identities. Using these results, we provide various criteria for the global existence and finite-time blow-up for the problem in the energy-subcritical regime. Finally, we establish a key compactness result, which enables us to obtain a characterization of finite-time blow-up solutions with minimal mass.

Nonlinear Schrödinger equations with critical Hardy potential and Choquard nonlinearity

Abstract

We study the Cauchy problem for the nonlinear Schrödinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Poho\v zaev identities. Using these results, we provide various criteria for the global existence and finite-time blow-up for the problem in the energy-subcritical regime. Finally, we establish a key compactness result, which enables us to obtain a characterization of finite-time blow-up solutions with minimal mass.

Paper Structure

This paper contains 14 sections, 19 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Hardy-Gagliardo-Nirenberg inequality and ground state solutions
  4. Hardy-Gagliardo-Nirenberg inequality
  5. Pohozǎev identities
  6. Blow-up and decay estimates
  7. Compactness of minimizing sequences and representation for minimizers
  8. Compactness of minimizing sequences
  9. Minimizers for the Hardy-Gagliardo-Nirenberg inequality
  10. Global existence and finite-time blow-up
  11. Global Existence
  12. Finite-time blow-up
  13. Minimal mass blow-up solutions
  14. Some auxiliary results

Key Result

Theorem 1.1

Assume $0<\alpha<d$ and $p>1$. Consider the ground state equation 1. Existence and Pohozǎev identities. Assume $\frac{d+\alpha}{d}<p<\frac{d+\alpha}{d-2}$. Then equation eq:EL-u admits a positive radial solution in $\mathcal{Q}_{\mu_0}$ which is monotonically decreasing in $|x|$. Moreover, any solution $w \in \mathcal{Q}_{\mu_0}$ to equation eq:EL-u satisfies the 2. Nonexistence. If $1<p \leq \

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2: Hardy-Gagliardo-Nirenberg inequality
  • Definition 1: Weak solutions
  • Theorem 1.3
  • Theorem 1.4: Minimal mass blow-up solutions
  • Lemma 2.1: Hardy-Littlewood-Sobolev inequality
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 27 more