ON STRONGLY QUASILINEAR DEGENERATE ELLIPTIC SYSTEMS WITH WEAK MONOTONICITY AND NONLINEAR PHYSICAL DATA

Abstract

This work is devoted to studying the quasilinear elliptic system $$\begin{aligned} -div ~ a(x,u,Du) + \vert u\vert ^{p-2} u +b(x,u,Du)=v(x)+f(x,u)+div ~g(x,u) \end{aligned}$$on a bounded open domain of \(\mathbb {R}^n\) with homogeneous Dirichlet boundary conditions. We show that there is a weak solution to this system under regularity, growth, and coercivity conditions for a, but only with very moderate monotonicity assumptions. We prove the existence result by using Galerkin’s approximation and the theory of Young measures.