On the dirichlet problem for a class of quasilinear elliptic systems of partial differential equations in divergence form

Abstract

(1.2) u(x) =uo(u), xEao where 111 : 0 C R t---+ RN is a vector valued function, i = 1,N, N 1, 0 is a bounded Lipschitz domain with boundary ao. (We use the summation convention throughout with i, i running from 1 to N and a, {3 running from 1 to n). We study the existence problem of weak solution of (1.1) • (1.2). A feature of our paper is that we treat a class of problems which the classical monotone operator methods developed by Visik [191, Minty (121, Browder [51, Bresis [21, Lions [101 do not work. The study of quasimonotone mappings is not only of the interest for function theory but also for its applications. For example, in the mathematical theory of nonlinear elastostatics, equations governing the equilibrium state of general homogeneous elastic materials without external forces are