On the existence of vector solutions to nonlinear Schr\"odinger equations with weak three-wave interaction

We study a nonlinear Schr\"odinger system with three-wave interaction: \begin{equation*} \left\{\begin{aligned} & - \Delta u_1 = f_1(u_1) + \alpha u_2u_3 \quad \text{ in } \R^N, & - \Delta u_2 = f_2(u_2) + \alpha u_3u_1 \quad \text{ in } \R^N, & - \Delta u_3 = f_3(u_3) + \alpha u_1u_2 \quad \text{ in } \R^N, & \quad \vec{u}=(u_1,u_2,u_3)\in (H_{\rm rad}^1(\R^N))^3, \end{aligned}\right. \end{equation*} where $3\leq N\leq 5$, $\alpha\in \R$ and each nonlinearity $f_i(\xi)$ satisfies the Berestycki-Lions conditions. Let $S_i$ denote the set of all least energy solutions of the scalar equation $-\Delta u = f_i(u)$ in $H_{\rm rad}^1(\R^N)$. A solution of the systems is called vector if all its components are nontrivial. We establish the existence of two distinct families of vector solutions $\{\vec{u}_\alpha\}$ with different asymptotic behaviors as $\alpha \to 0$. One family satisfies ${\rm dist}(\vec{u}_{\alpha},S_1\times S_2\times S_3) \to 0$, while another satisfies ${\rm dist}(\vec{u}_{\alpha},S_1\times S_2\times \{0\}) \to 0$. By contrast, we prove that no family of vector solutions satisfies ${\rm dist}(\vec{u}_{\alpha},S_1\times \{0\}\times \{0\}) \to 0$. Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.