In this paper, we consider the following nonlinear Schrödinger system with three wave interaction: \(\begin{aligned} {\left\{ \begin{array}{ll} - \varepsilon ^2 \Delta u_1 + V_1(x) u_1 = \mu _1 |u_1|^{p-1} u_1 + \alpha u_2 u_3,\quad \text {in}\ \mathbb {R}^N,\\ - \varepsilon ^2 \Delta u_2 + V_2(x) u_2 = \mu _2 |u_2|^{p-1} u_2 + \alpha u_1 u_3,\quad \text {in}\ \mathbb {R}^N,\\ - \varepsilon ^2 \Delta u_3 + V_3(x) u_3 = \mu _3 |u_3|^{p-1} u_3 + \alpha u_1 u_2,\quad \text {in}\ \mathbb {R}^N, \end{array}\right. } \end{aligned}\) where \(N \le 5\) , \(1< p < 2^* - 1\) , \(2^* = \infty \ (N \le 2)\) , \(2^* = 2N/(N-2)\ (N \ge 3)\) , \(\varepsilon > 0\) , \(V_j(x)>0\) , \(\mu _j > 0\ (j=1,2,3)\) and \(\alpha > 0\) . We construct a peak solution that is concentrating at a local minimum point of a function \(c(V_1(x),V_2(x),V_3(x))\) . Here \(c(\lambda _1,\lambda _2,\lambda _3)\) is a mountain pass value of the following limit system \(\begin{aligned} {\left\{ \begin{array}{ll} - \Delta v_1 + \lambda _1 v_1 = \mu _1 |v_1|^{p-1} v_1 + \alpha v_2 v_3\quad \text {in}\ \mathbb {R}^N,\\ - \Delta v_2 + \lambda _2 v_2 = \mu _2 |v_2|^{p-1} v_2 + \alpha v_1 v_3\quad \text {in}\ \mathbb {R}^N,\\ - \Delta v_3 + \lambda _3 v_3 = \mu _3 |v_3|^{p-1} v_3 + \alpha v_1 v_2\quad \text {in}\ \mathbb {R}^N. \end{array}\right. } \end{aligned}\) When \(p\in (1,2)\) , this limit system does not necessarily have a ground state. Hence a key of the construction is to use a local mountain pass approach.
