Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

Francesca Alessio; Piero Montecchiari; Andrea Sfecci; Francesca Alessio; Piero Montecchiari; Andrea Sfecci

doi:10.3934/nhm.2019022

Networks and Heterogeneous Media

2019, Volume 14, Issue 3: 567-587. doi: 10.3934/nhm.2019022

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Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

1.
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy
2.
Dipartimento di Ingegneria Civile Edile e Architettura, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy
3.
Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via Valerio 12/1, I-34127, Trieste, Italy

Received: 01 July 2018 Revised: 01 February 2019
35J50, 35J47, 34C37, 34C25, 35B40

We study systems of elliptic equations $ -\Delta u(x)+F_{u}(x, u) = 0 $ with potentials $ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $ which are periodic and even in all their variables. We show that if $ F(x, u) $ has flip symmetry with respect to two of the components of $ x $ and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on $ {\mathbb{R}}^{n} $.
- Saddle solutions,
- reversible elliptic equations,
- renormalized functional,
- variational methods
Citation: Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations[J]. Networks and Heterogeneous Media, 2019, 14(3): 567-587. doi: 10.3934/nhm.2019022

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Abstract

We study systems of elliptic equations $ -\Delta u(x)+F_{u}(x, u) = 0 $ with potentials $ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $ which are periodic and even in all their variables. We show that if $ F(x, u) $ has flip symmetry with respect to two of the components of $ x $ and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on $ {\mathbb{R}}^{n} $.

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