Singular perturbations of a class of elliptic variational–hemivariational inequalities without monotonicity

The aim of this article is to study the singular perturbations for a class of elliptic variational–hemivariational inequalities without strong monotonicity and relaxed monotonicity hypotheses. First, we prove existence of solutions to the original variational–hemivariational inequality under consideration, and its singular perturbation problem by applying an existence theorem for nonlinear equilibrium problems. Then, we provide a convergence result stating that the Kuratowski weak-upper limit of solution sets to singular perturbation problem coincides with the Kuratowski strong-upper limit of solution set to singular perturbation problem, and it is a nonempty subset of the solution set of original variational–hemivariational inequality. The singular perturbation analysis is completely new in the literature. Finally, in order to demonstrate the applicability of the results, an obstacle elliptic inclusion problem with convex subdifferential term, $p$-Laplace operator, and generalized Clarke’s subdifferential operator, is studied.