  We consider a class of equations in divergence form with a
singular/degenerate weight $$ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a
f(x,y)+\textrm{div}(|y|^aF(x,y))\;. $$ Under suitable regularity assumptions
for the matrix $A$, the forcing term $f$ and the field $F$, we prove H\"older
continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of
their derivatives. In addition, we show stability of the $C^{0,\alpha}$ and
$C^{1,\alpha}$ a priori bounds for approximating problems in the form $$
-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla
u)=(\varepsilon^2+y^2)^{a/2}
f(x,y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y)) $$ as $\varepsilon\to 0$.
Our method is based upon blow-up and appropriate Liouville type theorems.
