On the existence of positive solutions for certainquasilinear elliptic equations with Dirichlet boundary conditions
Z.-Q. Chen, R. J. Williams and Z. Zhao
We give sufficient conditions for the existence of positive solutions to some quasilinear elliptic equations in bounded domains $D \subset \mathbb R^n$ $(n\ge 3)$ with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form ${1\over 2} \Delta + b\cdot \nabla$ on $D$ are comparable when one modifies the drift term $b$ on a compact subset of $D$. This generalizes a previous result of Ancona, obtained under an $L^p$ condition on $b$, to a Kato condition on $|b|^2$.