Multiple normalized solutions for a competing system of Schrödinger equations

Abstract

We prove the existence of infinitely many solutions $λ_1, λ_2 \in \mathbb{R}$, $u,v \in H^1(\mathbb{R}^3)$, for the nonlinear Schrödinger system \[ \begin{cases} -Δu - λ_1 u = μu^3+ βu v^2 & \text{in $\mathbb{R}^3$} -Δv- λ_2 v = μv^3 +βu^2 v & \text{in $\mathbb{R}^3$} u,v>0 & \text{in $\mathbb{R}^3$} \int_{\mathbb{R}^3} u^2 = a^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a^2, \end{cases} \] where $a,μ>0$ and $β\le -μ$ are prescribed. Our solutions satisfy $u\ne v$ so they do not come from a scalar equation.