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In this paper, the aim of our work is to establish global weighted gradientestimates via fractional maximal functions and the point-wise regularityestimates 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*} thatrelated to Riesz potentials. Here, in our setting, $\Omega \subset\mathbb{R}^n$, $n \ge 2$ is a bounded Reifenberg flat domain (that its boundaryis sufficiently flat in sense of Reifenberg) and the small-BMO condition (smallbounded mean oscillations) is assumed on the nonlinearity $A$. Further, theemphasis of the paper is the existence of weak solution to a class ofquasilinear elliptic equations containing Riesz potential of the gradient term,as an application of the global point-wise bound. And regarding this study, wealso analyze the necessary and sufficient conditions that guarantee theexistence of solution to such nonlinear elliptic problems.

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