Multiple normalized solutions to a system of nonlinear Schrödinger equations

Abstract

We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{array}{l} -Δu_i+ λ_i u_i = \sum_{j=1}^Kβ_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2} \quad \mathrm{in} \, \mathbb{R}^3,\newline u_i \in H^1_{rad}(\mathbb{R}^3),\newline \int_{\mathbb{R}^3} |u_i|^2 \, dx = ρ_i^2 \quad \text{for }i=1,\ldots, K, \end{array} \right. \end{equation*} where $ρ=(ρ_1,\ldots,ρ_K)\in(0,\infty)^K$ is prescribed, $(λ,u) \in \mathbb{R}^K\times H^1(\mathbb{R}^3)^K$ are the unknown and $4\leq p<6$. In the case of two equations we show the existence of multiple solutions provided that the coupling is sufficiently large. We also show that for negative coupling there are no ground state solutions. The main novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in order to estimate the Morse index of a solution as well as a Liouville-type result in an exterior domain.