Existence results for quasilinear elliptic equations with multivalued nonlinear terms

In this paper we study the existence of solutions u ∈ ({{W}^{1,p}_{0}}) (Ω) with △ _p u ∈ L ²(Ω) for the Dirichlet problem 1 $$ left{ begin{array} [c]{l}-triangle_{p}uleft( xright) in-partial{Phi}left( uleft( xright) right) +Gleft( x,uleft( xright) right) ,xin{Omega}, umid_{partial{Omega}}=0, end{array} right. $$ where Ω ? R ^N is a bounded open set with boundary ?Ω, △ _p stands for the p?Laplace differential operator, ?Φ denotes the subdifferential (in the sense of convex analysis) of a proper convex and lower semicontinuous function Φ and G : Ω × R → 2^R is a multivalued map. We prove two existence results: the first one deals with the case where the multivalued map u ? G(x, u) is upper semicontinuous with closed convex values and the second one deals with the case when u ? G(x, u) is lower semicontinuous with closed (not necessarily convex) values.