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A tale of two approaches to heteroclinic solutions for Φ-Laplacian systems

Part of: Ordinary differential operators Equations and systems of special type Existence theories

Published online by Cambridge University Press:  14 May 2019

Yuan L. Ruan
Yuan L. Ruan*
Affiliation:
Beijing University of Aeronautics and Astronautics, 100191, Beijing, China (ruanyl@buaa.edu.cn)
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Abstract

In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory.

Keywords

heteroclinic solutionHamiltonian systemlocal minimizergeodesiclength metric space

MSC classification

Primary: 49J27: Problems in abstract spaces
Secondary: 35M99: None of the above, but in this section 34L30: Nonlinear ordinary differential operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2535 - 2572
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Copyright
Copyright © Royal Society of Edinburgh 2019

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