I am going through a proof of a Caccoppoli inequality for a PDE of the form
$$\sum_{|\alpha|, |\beta| \leqslant m} D^{\beta} \left( (-1)^{|\beta|} a_{\alpha\beta} D^{\alpha} u \right) = \sum_{|\beta| \leqslant m} (-1)^{|\beta|} D^{\beta} f_{\beta},$$ where $f_{\beta} \in L^2_{loc}(\Omega)$, $a_{\alpha\beta} \in L^{\infty}(\Omega)$ and is uniformly elliptic, i.e. $a_{\alpha\beta} \zeta_{\alpha}\zeta_{\beta} \geqslant \lambda |\zeta|^2$. We consider a weak solution $u \in H^m_{loc}(\Omega)$. One step in the proof involves the following implication.
If for all balls $B_{\rho'} \subset B_{\rho} \subset \Omega$ the estimate $$||u||_{H^m(B_{\rho'})} \leqslant \epsilon ||u||_{H^m(B_{\rho})} + C(\epsilon, \rho, \rho') \left( ||u||_{L^2(B_{\rho})} + ||\sum_{\beta} |f_{\beta}| ||_{L^2(B_{\rho})} \right)$$ is satisfied, then there exists $\tilde{C}(\epsilon, \theta)$ such that for each $\theta \in (0,1)$ and $\rho<1$ we have \begin{equation} (1) \qquad \qquad ||u||_{H^m(B_{\theta \rho})} \leqslant \epsilon ||u||_{H^m(B_{\rho})} + \frac{\tilde{C}(\epsilon, \theta)}{\rho^m} \left( ||u||_{L^2(B_{\rho})} + ||\sum_{\beta} |f_{\beta}| ||_{L^2(B_{\rho})} \right).\end{equation} It is suggested that this should be proved by a scaling argument. If I set $v(x) = u(\rho x)$ and $g(x) = f(\rho x)$ and use the norm on $H^m$ given by $$||\psi||_{H^m} = ||D^m \psi ||_{L^2} + ||\psi||_{L^2},$$ I can get (1) for $v(x) = u(\rho x)$ and $f$ replaced by $g$. My issue is that $v$ doesn't satisfy the same equation as $u$, and also the source function is now different. Is this even how one performs a "scaling argument"?
