EXISTENCE THEOREMS FOR ELLIPTIC HEMIVARIATIONAL INEQUALITIES INVOLVING THE p-LAPLACIAN

Here, 2 ≤ p < ∞, j : Z × R → R is a function which is measurable in z ∈ Z and locally Lipschitz in x ∈ R and ∂ j(z,x) is the Clarke subdifferential of j(z, ·). If f : Z × R → R is a measurable function which is in general discontinuous in the x ∈ R variable, for almost all z ∈ Z, all M > 0, and all |x| ≤ M, we have | f (z,x)| ≤ aM(z) with aM ∈ L1(Z) and we set j(z,x) = ∫x 0 f (z, r)dr, then j(z,x) is measurable in z ∈ Z, locally Lipschitz in x ∈ R and ∂ j(z,x) ⊆ [ f1(z,x), f2(z,x)] where f1(z,x) = liminfx′→x f (z,x′) and f2(z,x) = limsupx′→x f (z,x ′) (see Chang [5] and Clarke [6]). So problem (1.1) incorporates as a special case quasilinear elliptic problems with a discontinuous right-hand side which were studied by Chang [5]. Hemivariational inequalities are new type of inequality problems, which arise in mechanics and engineering when we wish to consider more realistic laws of nonmonotone and multivalued nature. This leads to energy functionals which are nonsmooth and nonconvex, and so the tools of differential calculus and convex analysis (which are used in the study of variational inequalities) are no longer suitable and new techniques based on the nonconvex and the