Weak solvability for a class of double phase variable exponents inclusion problems

We consider a large class of variable exponents double phase differential inclusions with mixed boundary conditions in a bounded domain with Lipschitz boundary. The motivation behind studying this problem is that it may be used in modelling the antiplane shear problem of a long cylinder, made of an anisotropic nonlinear Hencky-type material, in contact with a rigid obstacle. We derive a variational formulation in terms of Lagrange multipliers which formulates to a coupled system consisting of a double hemivariational inequality and a variational inequality. We introduce the corresponding Lagrange functional and show that any critical point, in the sense of Nonsmooth Analysis, of the Lagrangian corresponds to a weak solution of the problem under consideration. Existence and multiplicity results are then established via nonsmooth critical point theory.