A construction of peak solutions by a local mountain pass approach for a nonlinear Schrödinger system with three wave interaction

In this paper, we consider the following nonlinear Schrödinger system with three wave interaction: $$\begin{aligned} {\left\{ \begin{array}{ll} - \varepsilon ^2 \Delta u_1 + V_1(x) u_1 = \mu _1 |u_1|^{p-1} u_1 + \alpha u_2 u_3,\quad \text {in}\ \mathbb {R}^N,\\ - \varepsilon ^2 \Delta u_2 + V_2(x) u_2 = \mu _2 |u_2|^{p-1} u_2 + \alpha u_1 u_3,\quad \text {in}\ \mathbb {R}^N,\\ - \varepsilon ^2 \Delta u_3 + V_3(x) u_3 = \mu _3 |u_3|^{p-1} u_3 + \alpha u_1 u_2,\quad \text {in}\ \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ - ε 2 Δ u 1 + V 1 ( x ) u 1 = μ 1 | u 1 | p - 1 u 1 + α u 2 u 3 , in R N , - ε 2 Δ u 2 + V 2 ( x ) u 2 = μ 2 | u 2 | p - 1 u 2 + α u 1 u 3 , in R N , - ε 2 Δ u 3 + V 3 ( x ) u 3 = μ 3 | u 3 | p - 1 u 3 + α u 1 u 2 , in R N , where $$N \le 5$$ N ≤ 5 , $$1 0$$ ε > 0 , $$V_j(x)>0$$ V j ( x ) > 0 , $$\mu _j > 0\ (j=1,2,3)$$ μ j > 0 ( j = 1 , 2 , 3 ) and $$\alpha > 0$$ α > 0 . We construct a peak solution that is concentrating at a local minimum point of a function $$c(V_1(x),V_2(x),V_3(x))$$ c ( V 1 ( x ) , V 2 ( x ) , V 3 ( x ) ) . Here $$c(\lambda _1,\lambda _2,\lambda _3)$$ c ( λ 1 , λ 2 , λ 3 ) is a mountain pass value of the following limit system $$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta v_1 + \lambda _1 v_1 = \mu _1 |v_1|^{p-1} v_1 + \alpha v_2 v_3\quad \text {in}\ \mathbb {R}^N,\\ - \Delta v_2 + \lambda _2 v_2 = \mu _2 |v_2|^{p-1} v_2 + \alpha v_1 v_3\quad \text {in}\ \mathbb {R}^N,\\ - \Delta v_3 + \lambda _3 v_3 = \mu _3 |v_3|^{p-1} v_3 + \alpha v_1 v_2\quad \text {in}\ \mathbb {R}^N. \end{array}\right. } \end{aligned}$$ - Δ v 1 + λ 1 v 1 = μ 1 | v 1 | p - 1 v 1 + α v 2 v 3 in R N , - Δ v 2 + λ 2 v 2 = μ 2 | v 2 | p - 1 v 2 + α v 1 v 3 in R N , - Δ v 3 + λ 3 v 3 = μ 3 | v 3 | p - 1 v 3 + α v 1 v 2 in R N . When $$p\in (1,2)$$ p ∈ ( 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.