A priori bounds for a class of semi-linear degenerate elliptic equations

Abstract

In this paper, we mainly discuss a priori bounds of the following degenerate elliptic equation, {equation}\label{000} a^{ij}(x)\partial_{ij}u+b^i(x)\partial_i u +f(x,u)=0,\text{in}\Omega\subset\subset R^n, {equation} where $a^{ij}\partial_i \phi\partial_j \phi=0$ on $\partial \Omega$, $\phi$ is the defining function of $\partial \Omega$. Imposing suitable conditions on the coefficients and $f(x,u)$, one can get the $L^\infty$-estimates of \eqref{000} via blow up method.