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

TL;DR

This paper establishes a priori bounds for a class of degenerate elliptic equations, providing $L^ fty$ estimates under certain conditions using blow-up methods, which advances understanding of solution behavior in degenerate elliptic PDEs.

Contribution

It introduces new a priori bounds for semi-linear degenerate elliptic equations, expanding the theoretical framework for analyzing solutions with degeneracy on the boundary.

Findings

Derived $L^ fty$-estimates for solutions under specific coefficient conditions.

Applied blow-up method to establish bounds for degenerate elliptic equations.

Provided conditions ensuring boundedness of solutions in the domain.

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.