We are concerned with the following Schrödinger system with coupled quadratic nonlinearity −ε2Δv+P(x)v=μvw,x∈RN,−ε2Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN, $$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$ which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P ( x ), Q ( x ) are positive function potentials. By applying reduction method, we prove that if x 0 is a non-degenerate critical point of Δ ( P + Q ) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution ( v ε , w ε ) concentrating at x 0 for ε small enough.

中文翻译：

---

具有耦合二次非线性的非线性薛定谔系统的单峰解的存在性

我们关注以下具有耦合二次非线性的薛定谔系统 -ε2Δv+P(x)v=μvw,x∈RN,-ε2Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w> 0,v,w∈H1RN, $$\begin{方程}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu vw, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^ {2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^ {N}\right),\end{数组}\right. \end{equation}$$ 产生于二次介质中的二次谐波。这里 ε > 0 是一个小参数，2 ≤ N < 6, μ > 0 和 μ > γ, P ( x ), Q ( x ) 是正函数势。通过应用归约法，我们证明如果 x 0 是某个闭合的 N-1 维超曲面上的 Δ ( P + Q ) 的非退化临界点，