Entropy solutions for Leray-Lions anisotropic elliptic problem via the L¹ version of Minty's lemma

Abstract

We consider a class of nonlinear anisotropic elliptic equations with nonlinearities and measure data, governed by a Leray-Lions type operator associated with a Caratheodory function satisfying the natural growth condition and the coercivity condition, but it only satisfies the large monotonicity condition. We address the problem of the existence of entropy solutions in variable exponent anisotropic Sobolev spaces defined on a bounded domain. To overcome the difficulties arising from the lack of strict monotonicity, our existence theorem is proved by using the L¹-version of Minty's lemma.