Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

Abstract

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y{|}^{a}A(x,y)\nabla u)=|y{|}^{a}f(x,y)+\text{div}(|y{|}^{a}F(x,y)).$$ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder 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 $$-\mathrm{div}(({\epsilon }^2+y^2{)}^{a/2}A(x,y)\nabla u)=({\epsilon }^2+y^2{)}^{a/2}f(x,y)+\text{div}(({\epsilon }^2+y^2{)}^{a/2}F(x,y))$$ as $\epsilon \to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

48 pages