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 \[ \begin{cases} -Δu - λ_1 u = μu^3+ βu v^2 & \text{in ${\mathbb{R}}^3$} -Δv- λ_2 v = μv^3 +βu^2 v & \text{in ${\mathbb{R}}^3$} u,v>0 & \text{in ${\mathbb{R}}^3$} \int_{\mathbb{R}^3} u^2 = a^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a^2, \end{cases} \] where $a,\mu >0$ and $\beta \le -\mu$ are prescribed. Our solutions satisfy $u\ne v$ so they do not come from a scalar equation.

21 pages, substantial difference with respect to the previous version, the section 3 is entirely new

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