We consider coupled variational inequality and variational equality, and a case where inequality is constrained by a function of a solution to the variational equality. We show the existence of a solution to the coupled system. The problem is related to the Bean critical-state model in type II superconductors with a thermal effect in a multi-connected domain. Mathematically, we are concerned with coupled variational inequality and variational equality containing a p-curl inequality constraint by a function of the solution to q-Laplacian equality.

quasivariational inequality, stationary coupled system, weak solution.