Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation
$$
\Delta u-V(x)u=0,
$$
where $V$ is a bounded function. If we require that $\int_{\Omega_{\rho}}{|u|^2w(x)dx}<\infty$, here $w(x)$ is some weight function, for example, we choose $w(x)=\exp{|x|^{\alpha}}$, $\alpha>0$. Then my question is how to show that 
$$
\int_{\Omega_{\rho}}{|D^{\beta}u|^2w(x)dx}<\infty
$$
is also true for $|\beta|\leq 2$?

Thanks in advance.