# Linking Methods for Componentwise Variational Systems

## Abstract

The paper deals with the equilibrium solutions of two-equation systems in which each component equation has a variational structure. Solutions are obtained that can be in one of the following generalized Nash situations: (a) one component of the solution represents a mountain pass type critical point and the other is a minimizer; (b) both components of the solution are mountain pass type; (c) both components are minimizers, that is, the solution is a proper Nash equilibrium. The simultaneous treatment of critical points of the mountain pass type and of the minimum ones is achieved by using a unifying notion of linking. The theory is applied to a system of four elliptic equations in which the subsystems formed by the first two and the last two equations, respectively, are of gradient type. An example shows that the conditions found are non-contradictory. The theory could be applied to other classes of systems.

## Authors

Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, Cluj-Napoca, Romania Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Andrei Stan
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania

## Keywords

Variational method; linking; critical point; mountain pass geometry; Nash type equilibrium; monotone operator; elliptic system

## Paper coordinates

R. Precup, A. Stan, Linking methods for componentwise variational systems, Results Math. 78 (2023) 246, https://doi.org/10.1007/s00025-023-02026-x

## PDF

??

##### Journal

Results in Mathematics

Springer

1422-6383
##### Online ISSN

1420-9012

 Ambrosetti, A., Rabinowitz, P.H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
 Beldinski, M., Galewski, M., Nash type equilibria for systems of non-potential equations. Appl. Math. Comput. 385, 125456 (2020)
 Benci, V., Rabinowitz, P.H., Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979)
 Boureanu, M.-M., Pucci, P., Radulescu, V.D., Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent. Complex Var. Elliptic Equ. 56, 755–767 (2011)
 Chabrowski, J., Introduction to the theory of critical points. The mountain pass theorem. Ekeland’s variational principle. In: Instructional Workshop on Analysis and Geometry, Part III, Canberra (1995)
 Costa, D.G., Magalhaes, C.A., Existence results for perturbations of the p -Laplacian. Nonlinear Anal. 24, 409–418 (1995)
 Costea, N., Csirik, M., Varga, C., Linking-type results in nonsmooth critical point theory and applications. Set-Valued Var. Anal. 25, 333–356 (2017)
 De Figueiredo, D.G., Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research, Bombay (1989)
 Filippucci, R., Pucci, P., Robert, F., On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. 91(2), 156–177 (2009)
 Galewski, M.,  On the mountain pass solutions to boundary value problems on the Sierpinski gasket. Results Math. 74, 167 (2019)
 Grossinho, M.R., Tersian, S.A., An Introduction to Minimax Theorems and Their Applications to Differential Equations. Springer, Dordrecht (2001)
 Jebelean, P., Moro¸sanu, Gh., Mountain pass type solutions for discontinuous perturbations of the vector p-Laplacian. Nonlinear Funct. Anal. Appl. 10(4), 591–611 (2005)
 Kassay, G., R˘adulescu, V.D., Equilibrium Problems and Applications. Academic Press (2019)
 Le Dret, H., Nonlinear Elliptic Partial Differential Equations. Springer, Berlin (2018)
 Motreanu, D., Motreanu, V.V., Papageorgiou, N.S., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
 Mugnai, D., Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem. NoDEA Nonlinear Differ. Equ. Appl. 11, 379–391 (2004)
 Precup, R., Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal. 75, 834–851 (2012)
 Precup, R., Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3(4), 197–207 (2014)
 Precup, R., A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems. J. Math. Anal. Appl. 463, 412–431 (2018)
 Precup, R., Componentwise localization of critical points for functionals defined on product spaces. Topol. Methods Nonlinear Anal. 58, 51–77 (2021)
 Precup, R., Stan, A., Stationary Kirchhoff equations and systems with reaction terms. AIMS Math. 7(8), 15258–15281 (2022)
 Pucci, P., Radulescu, V., The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 9(3), 543–584 (2010)
 Rabinowitz, P.H., Minimax methods in critical point theory with applications to nonlinear partial differential equations. Conf. Board of Math Sci. 65, Amer. Math. Soc. (1986)
 Schechter, M., Linking Methods in Critical Point Theory. Birkhauser, Boston (1999)
 Silva, E.A.B., Existence and multiplicity of solutions for semilinear elliptic systems. Nonlinear Differ. Equ. Appl. 1, 339–363 (1994)
 Stan, A., Nonlinear systems with a partial Nash type equilibrium. Studia Univ. Babes-Bolyai Math. 66, 397–408 (2021)
 Struwe, M., Variational Methods. Springer, Berlin (1990)