Quasilinear Elliptic Equations via Gradient-Form-Boundedness

Abstract

This paper studies the Dirichlet problem for quasilinear elliptic equations with singular coefficients, extending the classical Ladyzhenskaya-Ural'tseva theory. We replace the standard L-r-integrability condition (with r greater than n) with a more flexible p-gradient-form-boundedness assumption. Under this weakened hypothesis, we establish local and global a priori estimates, existence of weak solutions, and Holder continuity of bounded weak solutions. The results hold on arbitrary bounded open sets without boundary regularity requirements.