On Carlson's Type Removability Test for the Degenerate Quasilinear Elliptic Equations

Abstract

Carlson's type theorem on removable sets for $\alpha$-Holder continuous solutions is investigated for the quasilinear elliptic equations $\text{div}\hspace{0.17em}A(x,u,\nabla u)=0,$ having degeneration $\omega$ in the Muckenhoupt class. In partial, when $\alpha$ is sufficiently small and the operator is weighted $p$-Laplacian, we show that the compact set $E$ is removable if and only if the Hausdorff measure ${\Lambda }_{\omega }^{-p+\left(p-1\right)\alpha }\left(E\right)=0$.