Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

We consider a class of equations in divergence form with a singular/degenerate weight (Formula presented.) Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Hölder continuity of solutions which are even in (Formula presented.) and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the (Formula presented.) and (Formula presented.) a priori bounds for approximating problems in the form (Formula presented.) as (Formula presented.) Finally, we derive (Formula presented.) and (Formula presented.) bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.