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

Source abstract

<abstract><p>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 <italic>local boundedness</italic> 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.</p></abstract>