# Prescribed mass ground states for a doubly nonlinear Schr\"odinger equation in dimension one

## TL;DR

This paper studies the existence and uniqueness of ground states with fixed mass for two types of focusing nonlinear Schrödinger equations in one dimension, revealing conditions under which ground states exist or are unique.

## Contribution

It provides new results on existence and uniqueness of ground states for doubly nonlinear Schrödinger equations, including critical and subcritical regimes, with detailed mass thresholds.

## Key findings

- Existence and uniqueness at all masses in subcritical regimes.
- Critical mass thresholds depend on nonlinearities.
- Ground states exist only at specific critical masses in the doubly critical case.

## Abstract

We investigate the problem of existence and uniqueness of ground states at fixed mass for two families of focusing nonlinear Schr\"odinger equations on the line. The first family consists of NLS with power nonlinearities concentrated at a point. For such model, we prove existence and uniqueness of ground states at every mass when the nonlinearity power is $L^2-$subcritical and at a threshold value of the mass in the $L^2-$critical regime. The second family is obtained by adding a standard power nonlinearity to the previous setting. In this case, we prove existence and uniqueness at every mass in the doubly subcritical case, namely when both the powers related to the pointwise and the standard nonlinearity are subcritical. If only one power is critical, then existence and uniqueness hold only at masses lower than the critical mass associated to the critical nonlinearity. Finally, in the doubly critical case ground states exist only at critical mass, whose value results from a non--trivial interplay between the two nonlinearities.