# Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case

## TL;DR

This paper investigates the existence and stability of normalized ground states for the Sobolev critical nonlinear Schrödinger equation with combined nonlinearities, filling a gap in the understanding of such solutions in the whole space.

## Contribution

It provides the first comprehensive analysis of normalized ground states for the Sobolev critical NLSE with combined nonlinearities in ^* case, including existence and stability results.

## Key findings

- Existence of ground states in certain parameter regimes.
- Non-existence or instability in other regimes.
- Extension of Brezis-Nirenberg problem to normalized solutions.

## Abstract

We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $\mu |u|^{q-2} u$ we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space $\mathbb{R}^N$.