Local boundedness of weak solutions to elliptic equations with p,q−growth

Abstract

This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.

Keywords

• regularity
• local boundedness
• weak solutions
• elliptic equations
• $ p， q- $growth