Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

Abstract: In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*} \mathrm{div}(A(x,\nabla u)) = \mathrm{div}(f) \ \text{in} \ \Omega, \mbox{ and } \ u = g \ \text{on} \ \partial \Omega, \end{align*} that related to Riesz potentials. Here, in our setting, $\Omega \subset \mathbb{R}^n$, $n \ge 2$ is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity $A$. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.