Nonlinear Schr\"{o}dinger equations with critical Hardy potential and Choquard nonlinearity

TL;DR

This paper investigates the nonlinear Schrödinger equation with a critical Hardy potential and Choquard nonlinearity, establishing existence, non-existence, and blow-up criteria through variational and compactness methods.

Contribution

It introduces new criteria for solution behavior and characterizes minimal mass blow-up solutions using a novel compactness result.

Findings

Existence of ground state solutions via Hardy-Gagliardo-Nirenberg optimizers

Non-existence of solutions using Pohozaev identities

Criteria for global existence and finite-time blow-up

Abstract

We study the Cauchy problem for the nonlinear Schr\"{o}dinger 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.