Existence and multiplicity for quasilinear elliptic inclusions with nonmonotone discontinuous multifunction

We develop a flexible tool in terms of appropriately defined sub- and supersolutions that allows for obtaining existence, bounds, and multiplicity of solutions for quasilinear elliptic inclusions with a discontinuous multi-function that need neither be upper nor lower semicontinuous. The problem under consideration may be used to describe certain free boundary problems in mechanical models involving multi-valued constitutive laws. Our approach is based on a combined use of abstract fixed point results for monotone mappings on partially ordered sets, and on the existence and comparison results for multi-valued quasilinear elliptic problems with Clarke’s generalized gradient.