A (p−q) coupled system in elliptic nonlinear problems with nonstandard boundary conditions

We state an abstract variational formulation to a coupled system constituted by an inequality and an equality motivated by the motion and energy equations, and the constitutive laws for the stress tensor and the heat flux, respectively, when non-Newtonian fluids are taken care of. Here the existence of a weak solution is proven via a fixed point argument to multivalued mappings. The nonstandard boundary conditions correspond to friction wall laws and energy transfer condition considered on a part of the boundary, whereas there exists the presence of the frictional work due to the friction of the fluid motion. We conclude by formulating the corresponding stationary heat conducting viscous incompressible flow problem and we establish an existence result.