Let $u$ be a solution (in the distributional sense) of $$ \Delta u = \delta_r $$ on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$. Let $w$ be a solution of $$ Aw = \delta_r $$ where $A = \partial_{x^i}(a^{ij}\partial_{x^j})$, with $a^{ij} \in C^\infty(\Omega)$, $a^{ij}(r) = Id$ and $(\partial_{x^k}a^{ij})(r)=0$ for all $i, j, k \in \{1,2 \}$. Does it follow that, in a neighborhood of $r$, $u-w$ is bounded?
One could go about it saying: let's find a uniform bound for the laplacian of $u-w$ outside of $r$ (by elliptic regularity, $u$ and $w$ are smooth outside of $r$), and then from this somehow deduce boundedness of $u-w$. But, for instance, $\log(|x-r|)$ is harmonic and unbounded, so this does not work. What am I missing?
Instead of $u-w$, I think it's easier to show that $u-\log|x-r|$ and $w-\log |x-r|$ are both bounded. For former this is clear because it's harmonic. For the latter, $$ \begin{split} A(-w+\log |x-r|)&=-\delta_r + \sum_{ij} (\partial_{x^i} a^{ij}) \partial_{x^j}\log|x-r| + \sum_{ij} a^{ij}\partial_{x^ix^j}\log|x-r| \\ &=\sum_{ij} (\partial_{x^i} a^{ij}) \partial_{x^j}\log|x-r| + \sum_{ij} (a^{ij}-\delta^{ij})\partial_{x^ix^j}\log|x-r| \end{split}$$ Here $\delta^{ij}$ is Kronecker's, not Dirac's.
The $n$th derivative of $\log $ is $O(|x-r|^{-n})$ but the terms $\partial_{x^i} a^{ij}$ and $(a^{ij}-\delta^{ij})$ decay fast enough to offset that. You get an equation of the form $A(-w+\log |x-r|)=f$ with reasonable $f$ (vanishing at $r$), from where the regularity of $-w+\log |x-r| $ follows.