# Ground state of the mass-critical inhomogeneous nonlinear Schrodinger functional

## TL;DR

This paper investigates the existence and behavior of ground states in a mass-critical inhomogeneous nonlinear Schrödinger functional, revealing conditions for existence and a universal blow-up profile in the critical regime.

## Contribution

It establishes the optimal conditions for ground state existence and characterizes the universal blow-up profile using a Gagliardo-Nirenberg inequality.

## Key findings

- Optimal existence conditions for ground states
- Universal blow-up profile in the critical regime
- Connection to Gagliardo-Nirenberg inequality

## Abstract

We study the ground state problem of the nonlinear Schrodinger functional with a mass-critical inhomogeneous nonlinear term. We provide the optimal condition for the existence of ground states and show that in the critical focusing regime there is a universal blow-up profile given by the unique optimizer of a Gagliardo-Nirenberg interpolation inequality.