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 $|\beta|\leq 2$?

I came across this problem when reading a paper, and reference the author offered there was Hormander's The Analysis of Linear Differential operators, the 2nd volume, where I failed to find something similar. I'm very appreciated that if someone can point out the exact theorem or proposition that corresponds to the above result in Hormander's book. Thanks in advance.