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 the reference the author offered was Hormander's The Analysis of Linear Differential operators, the 2nd volume, where I failed to find something similar. However, I found that in the first section of Chapter 17, he gave the interior Shauder estimates by using fundamental solution and fractional integral. But the result is local and without weight. I don't know if this can modified to the case I concerned here. Another resource that maybe helpful is S Agmon, A Douglis, L Nirenberg's classical paper "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions.I" , in thm 15.1, they established the interior estimated for general elliptic operators, but I don't know whether their methods can be applied to the case with weight like $w(x)=\exp{|x|^{\alpha}}$. I'm very appreciated that if someone can point out how this is done. Thanks in advance.

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