NODAL Vector solutions with clustered peaks for a nonlinear elliptic equations in $\R^3$

In this paper, we study the following coupled nonlinear Schr\"{o}dinger system in $\R^3$ $$ \left\{% \begin{array}{ll} -\epsilon^2\Delta u +P(x)u=\mu_1 u^3+\beta v^2u,~~&x\in \R^3,\vspace{0.15cm}\\ -\epsilon^2\Delta v +Q(x)v=\mu_2 v^3+\beta u^2v,~~&x\in \R^3,\\ \end{array}% \right. $$ where $\mu_1 >0,\mu_2>0$ and $\beta \in \R$ is a coupling constant. Whether the system is repulsive or attractive, we prove that it has nodal semi-classical segregated or synchronized bound states with clustered spikes for sufficiently small $\epsilon$ under some additional conditions on $P(x), Q(x)$ and $\beta$. Moreover, the number of this type of solutions will go to infinity as $\epsilon \to 0^+$.
Comment: 36 pages