Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

Abstract

We study regularity for solutions of quasilinear elliptic equations of the form \mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F} in bounded domains in \mathbb{R}^n . The vector field \mathbf{A} is assumed to be continuous in u , and its growth in \nabla 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 \mathbf{A} . As a consequence, we obtain that \nabla u is in the classical Morrey space \mathcal{M}^{q,\lambda} or weighted space L^q_w whenever |\mathbf{F}|^{1/(p-1)} is respectively in \mathcal{M}^{q,\lambda} or L^q_w , where q is any number greater than p and w is any weight in the Muckenhoupt class A_{q/p} . In addition, our two-weight estimate allows the possibility to acquire the regularity for \nabla u in a weighted Morrey space that is different from the functional space that the data |\mathbf{F}|^{1/(p-1)} belongs to.

Locations

• Revista Matemática Iberoamericana - View
• arXiv (Cornell University) - View - PDF