Orbital stability of ground states for a Sobolev critical Schr\"odinger equation

TL;DR

This paper proves the orbital stability of ground states for a Sobolev critical nonlinear Schrödinger equation with mixed nonlinearities, establishing their characterization as local energy minima and answering a question posed by N. Soave.

Contribution

It demonstrates the orbital stability of ground states in a Sobolev critical setting, despite the critical nonlinearity, and characterizes these states as local energy minima.

Findings

Ground states are local minima of the energy functional.

The set of ground states is orbitally stable.

Results resolve a question raised by N. Soave.

Abstract

We study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities \begin{equation*} i \partial_t v + \Delta v + \mu v |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \end{equation*} where $N \geq 3$, $v: \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $\mu > 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, we show that the set of ground states is orbitally stable. Our results settle a question raised by N. Soave [35].