Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations.

We study regularity for solutions of quasilinear elliptic equations of the form divA(x, u,u) = divF in bounded domains in Rn. The vector field A is assumed to be continuous in u, and its growth in u is like that of the p-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions u to the equation under a small BMO condition in x for A. As a consequence, we obtain that u is in the classical Morrey space Mq, or weighted space Lq w whenever |F|1/(p-1) is respectively in Mq, or Lq w, where q is any number greater than p and w is any weight in the Muckenhoupt class Aq/p. In addition, our twoweight estimate allows the possibility to acquire the regularity for u in a weighted Morrey space that is different from the functional space that the data |F|1/(p-1) belongs to. [ABSTRACT FROM AUTHOR]