Multiple normalized solutions for a competing system of Schrödinger equations

Abstract

We prove the existence of infinitely many solutions $\lambda _1, \lambda _2 \in \mathbb {R}$, $u,v \in H^1(\mathbb {R}^3)$, for the nonlinear Schrödinger system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u - \lambda _1 u = \mu u^3+ \beta u v^2 &{} \text {in }\mathbb {R}^3 \\ -\Delta v- \lambda _2 v = \mu v^3 +\beta u^2 v &{} \text {in }\mathbb {R}^3\\ u,v>0 &{} \text {in }\mathbb {R}^3 \\ \int _{\mathbb {R}^3} u^2 = a^2 \quad \text {and} \quad \int _{\mathbb {R}^3} v^2 = a^2, \end{array}\right. } \end{aligned}$$

where $a,\mu >0$ and $\beta \le -\mu$ are prescribed. Our solutions satisfy $u\ne v$ so they do not come from a scalar equation. The proof is based on a new minimax argument, suited to deal with normalization conditions.