H\"older estimates and asymptotic behaviors of solutions of some degenerate elliptic equations on half-spaces

Abstract

In this paper we report the asymptotic behaviors of viscosity solutions of the following degenerate elliptic equations \begin{equation*}\label{main-Eq} Lu=x_n^{2\alpha}\sum_{i,j=1}^{n-1}a_{ij}(x)D_{ij}u(x) +2x_n^{\alpha}\sum_{i=1}^{n-1}a_{in}(x)D_{in}u(x)+D_{nn}u(x)=0 \end{equation*} outside a bounded domain of the upper half space for $\alpha>0$ with proper coefficients. Meanwhile, the H\"older estimates up to the boundary will be obtained using rescaling methods.