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}Ω\subset\subset R^n, {equation} where $a^{ij}{\partial }_i\varphi {\partial }_j\varphi =0$ on $\partial \Omega$, $\varphi$ 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.

17 pages