Existence for coupled pseudomonotone–strongly monotone systems and application to a Cahn–Hilliard model with elasticity

A system of two operator equations is considered–one of pseudomonotone type and the other of strongly monotone type–both being strongly coupled. Conditions are given that allow to reduce the solvability of this system to a single operator equation of a pseudomonotone mapping. This result is applied to a coupled system consisting of a parabolic equation of fourth order in space of Cahn–Hilliard type and a nonlinear elliptic equation of second order modeling a quasi-steady mechanical equilibrium. Using an appropriate notion of weak solutions and a framework for evolution equations developed by Gröger (2001), the system is reduced to a single parabolic operator equation and the existence of solutions is shown under restrictions on the strength of the coupling.