Abstract
Given a nonnegative, smooth potential \(V: {{\mathbb {R}}}^k \rightarrow {{\mathbb {R}}}\) (\(k \ge 2\)) with multiple zeros, we say that a curve \({\mathfrak {q}}: {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}^k\) is a connecting orbit if it solves the autonomous system of ordinary differential equations
and tends to a zero of V at \(\pm \infty \). Broadly, our goal is to study the existence of connecting orbits for the problem above using variational methods. Despite the rich previous literature concerning the existence of connecting orbits for other types of second order systems, to our knowledge only connecting orbits which minimize the associated energy functional in a suitable function space were proven to exist for autonomous multi-well potentials. The contribution of this paper is to provide, for a class of such potentials, some existence results regarding non-minimizing connecting orbits. Our results are closely related to the ones in the same spirit obtained by J. Bisgard in his PhD thesis (University of Wisconsin-Madison, 2005), where non-autonomous periodic multi-well potentials (ultimately excluding autonomous potentials) are considered. Our approach is based on several refined versions of the classical Mountain Pass Lemma.
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Notes
We say that J satisfies the Palais–Smale condition at the level \(c \ge {\mathfrak {m}}\) if every sequence satisfying (2.7) possesses a convergent subsequence in \({\mathscr {H}}\).
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Acknowledgements
I wish to thank my PhD advisor Fabrice Bethuel for bringing this problem into my attention and for many useful comments and remarks during the elaboration of this paper. I also wish to thank the referee for pointing to several important references such as [11] as well as for numerous remarks and suggestions which lead to significant improvements on the paper. This program has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No 754362.
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Oliver-Bonafoux, R. Non-minimizing connecting orbits for multi-well systems. Calc. Var. 61, 69 (2022). https://doi.org/10.1007/s00526-021-02167-3
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DOI: https://doi.org/10.1007/s00526-021-02167-3