On the existence of vector solutions to nonlinear Schrödinger equations with weak three-wave interaction

Abstract

We study a nonlinear Schrödinger system with three-wave interaction: \begin{equation*} \left\{\begin{aligned} & - Δu_1 = f_1(u_1) + αu_2u_3 \quad \text{ in } \R^N, & - Δu_2 = f_2(u_2) + αu_3u_1 \quad \text{ in } \R^N, & - Δu_3 = f_3(u_3) + αu_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 $3\le N\le 5$, $\alpha \in \mathbb{R}$ 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 $H_{\mathrm{rad}}^1({\mathbb{R}}^N)$. 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 $\mathrm{dist}({\vec{u}}_{\alpha },S_1\times S_2\times S_3)\to 0$, while another satisfies $\mathrm{dist}({\vec{u}}_{\alpha },S_1\times S_2\times \{0\})\to 0$. By contrast, we prove that no family of vector solutions satisfies $\mathrm{dist}({\vec{u}}_{\alpha },S_1\times \{0\}\times \{0\})\to 0$. Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.