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Applying techniques originally developed for systems lacking a variationalstructure, we establish conditions for the existence of solutions in systemsthat possess this property but their energy functional is unbounded both lowerand below. We show that, in general, our conditions differ from those in theclassical mountain pass approach by Ambroseti-Rabinovitz when dealing withsystems of this type. Our theory is put into practice in the context of acoupled system of Stokes equations with reaction terms, where we establishsufficient conditions for the existence of a solution. The systems under studyare intermediary between gradient-type systems and Hamiltonian systems

Role of partial functionals in the study of variational systems