Existence Results for Quasilinear Elliptic Equations with Multivalued Nonlinear Terms

Abstract

In this paper we study the existence of solutions u ∈ W1,p0(Ω) with △pu ∈ L2(Ω) for the Dirichlet problem {−△pu(x)∈−∂Φ(u(x))+G(x,u(x)),x∈Ω,u∣∂Ω=0, (1) where Ω ⊆ RN 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 → 2R 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.