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

We consider a class of equations in divergence form with a singular/degenerate weight -div([absolute value of y].sup.a]A(x, y)[nabla]u) = [[absolute value of y].sup.a]f(x, y) + div([[absolute value of y].sup.a]F(x, y)). Under suitable regularity assumptions for the matrix A, the forcing term f and the field F, we prove Holder continuity of solutions which are odd in y [member of] R, and possibly of their derivatives. In addition, we show stability of the [C.sup.0,[alpha]] and [C.sup.1,[alpha]] a priori bounds for approximating problems in the form -div([([[epsilon].sup.2] + [y.sub.2]).sup.a/2]A(x, y)[nabla]u) = [([[epsilon].sup.2] + [y.sub.2]).sup.a/2] f(x, y) + div([([[epsilon].sup.2] + [y.sub.2]).sup.a/2] F (x, y)) as [epsilon] [right arrow] 0. Our method is based upon blow-up and appropriate Liouville type theorems. Keywords: degenerate and singular elliptic equations; Liouville type theorems; blow-up; fractional Laplacian; divergence form elliptic operator; Schauder estimates; boundary Harnack; Fermi coordinates