Multi-scale spike solutions for nonlinear coupled elliptic systems with critical frequency

Abstract

In this paper, we study the following elliptic system

$$\begin{aligned} \left\{ \begin{array}{ll}-\varepsilon ^2\Delta u+P_1(x)u=\mu _1 u^3+\beta uv^2 &{}\text{ in } {\mathbb {R}}^3,\\ -\varepsilon ^2\Delta v+P_2(x)v=\mu _2 v^3+\beta u^2v &{}\text{ in } {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$

with critical frequency in the sense that $\inf _{x\in {\mathbb {R}}^N}P_i(x)=0(i=1,2)$. We show that if the coupling constant $\beta$ is relatively large and the zero set of $P_1(x)$ and $P_2(x)$ admits several common isolated connected components, there exist synchronized positive vector solutions which are trapped in a neighborhood of the zero set of $P_i(x)(i=1,2)$ and also the local maximum points of $P_i(x)(i=1,2)$ for $\varepsilon$ small. Moreover, the amplitudes of the solutions around the components of the zero set and local maximum points are of a different order of $\varepsilon$.