We have the following nonlinear Schrodinger equations ($n\leq3$):
$$\begin{cases} \Delta u_1 -u_1+\mu_1u_1^3+\beta u_1u_2^2=0\\ \Delta u_2 -\lambda u_2+\mu_2u_2^3+\beta u_1^2u_2=0\\ u_1,\,u_2\in H^1(\mathbb{R}^n)=W^{1,2}(\mathbb{R}^n) \end{cases}$$
How to use the classical "bootstrap" argument to show that the solution of the above system $u_1,\,u_2$ are in $C^2(\mathbb{R}^n)$ and tend to zero as $x\rightarrow \infty$?