Multiple normalized solutions for a Sobolev critical Schr\"odinger equation

We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities $$i \partial_t \phi + \Delta \phi + \mu \phi |\phi|^{q-2} + \phi |\phi|^{2^* - 2} = 0, \quad (t, x) \in R \times R^N,$$ where $N \geq 3$, $\phi: R \times R^N \to C$, $\mu > 0$, $2 < q < 2 + 4/N$ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. It was already proved that, for small mass, ground states exist and correspond to local minima of the associated Energy functional. It was also established that despite the nonlinearity is Sobolev critical, the set of ground states is orbitally stable. Here we prove that, when $N \geq 4$, there also exist standing waves which are not ground states and are located at a mountain-pass level of the Energy functional. These solutions are unstable by blow-up in finite time. Our study is motivated by a question raised by N. Soave.