Multiple normalized solutions for a Sobolev critical Schrödinger equation

Abstract

We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schrödinger equation with mixed power nonlinearities

$$\begin{aligned} i \partial _t \phi + \Delta \phi + \mu \phi |\phi |^{q-2} + \phi |\phi |^{2^* - 2} = 0, \quad (t, x) \in \mathbb {R}\times \mathbb {R}^N, \end{aligned}$$

where $N \ge 3$, $\phi : \mathbb {R}\times \mathbb {R}^N \rightarrow \mathbb {C}$, $\mu > 0$, $2< q < 2 + 4/N$ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. It was proved in Jeanjean et al. (Orbital stability of ground states for a Sobolev critical Schrödinger equation, 2020) 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 $N \ge 4$, 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 Soave (J Funct Anal 279(6):108610, 2020).