This is an extension of Casado-D¿az’s earlier work [1]. It is well known that the non homogenous Dirichlet problem:where is a pseudomonotone operator of the Leray-Lions type, defined in: and f is in L¹(&OHgr;), may not have a unique solution.

In this work, the authors prove the existence of maximal and minimal renormalized solutions, utilizing some suitable assumptions of locally Lipschitz or locally Holder, and continuity of a(x,s, &xgr;) with respect to s. The main results are given in Theorem 1.1 where it is shown that if a satisfies the authors’ conditions that are stated in equations (1.2) through (1.5), or in (1,2), (1.3), (1.6), and (1.7), then it is sufficient to have maximal and minimal renormalized solutions; and in theorem 2.6, where it is proved that if and u are renormalized solutions, then any other renormalized solution z will satisfy u < z < almost everywhere on &OHgr;.