Role of partial functionals in the study of variational systems


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


Andrei Stan
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca,  Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania


Variational method; Stokes system; Mountain pass geometry

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Andrei Stan, Role of partial functionals in the study of variational,


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[1] Ambrosetti, A., Rabinowitz, P.H., 1973. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381.
[2] Berman, A., Plemmons, R.J., 1979. Nonnegative Matrices in the Mathematical Sciences. Academic Press, Philadelphia.
[3] Brumar, D., 2023. A fixed point approach to the semi-linear Stokes problem. Studia Univ. Babe¸s-Bolyai Math. 68.
[4] De Figueiredo, D., 1989. Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research .
[5] Galdi, G.P., 2011. An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems. 2nd ed., Springer, New York.
[6] Girault, V., Raviart, P.A., 1986. Finite Element Methods for Navier–Stokes Equations. Springer, Berlin.
[7] Kohr, M., Precup, R., 2023. Analysis of Navier–Stokes Models for Flows in Bidisperse Porous Media. J. Math. Fluid Mech. 38.
[8] Precup, R., 2014. Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3, 197–207.
[9] Precup, R., Stan, A., 2023. Linking methods for componentwise variational systems. Results Math 78.
[10] Sohr, H., 2001. The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Springer Basel, Basel.
[11] Stan, A., 2021. Nonlinear systems with a partial Nash type equilibrium. Studia Univ. Babe¸s-Bolyai Math. 66, 397–408.
[12] Stan, A., 2023. Nash equilibria for componentwise variational systems. J.  Nonlinear Funct. Anal. 66.
[13] Teman, R. (Ed.), 2001. Navier–Stokes Equations. Theory and Numerical Analysis. AMS Chelsea edn. American Mathematical Society, UK edition.


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