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Linking Methods for Componentwise Variational Systems

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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.

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their reviews and valuable remarks, which significantly improved the paper.

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This research did not benefit from any kind of funding.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Radu Precup

  2. Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110, Cluj-Napoca, Romania

    Radu Precup & Andrei Stan

  3. Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Andrei Stan

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  1. Radu Precup
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  2. Andrei Stan
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Both authors contributed equally to the study, conception and design. Both authors read and approved the final manuscript.

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Correspondence to Radu Precup.

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Precup, R., Stan, A. Linking Methods for Componentwise Variational Systems. Results Math 78, 246 (2023). https://doi.org/10.1007/s00025-023-02026-x

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