Orbital stability of ground states for a Sobolev critical Schrödinger equation

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].