Homoclinic orbits of first-order superquadratic  Hamiltonian systems

Cyril Joel Batkam

doi:10.3934/dcds.2014.34.3353

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2014, Volume 34, Issue 9: 3353-3369. Doi: 10.3934/dcds.2014.34.3353

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Homoclinic orbits of first-order superquadratic Hamiltonian systems

Cyril Joel Batkam^1,

1.
Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1

Received: April 30, 2013

Revised: October 31, 2013

Published: August 2014

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Abstract

In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system \begin{equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}. \end{equation*} Under the Ambrosetti-Rabinowitz's superquadraticy condition, or no Ambrosetti-Rabinowitz's superquadracity condition, we present two results on the existence of infinitely many large energy homoclinic orbits when $H$ is even in $u$. We apply the generalized (variant) fountain theorems established recently by the author and Colin. Under no Ambrosetti-Rabinowitz's superquadracity condition, we also obtain the existence of a ground state homoclinic orbit by using the method of the generalized Nehari manifold for strongly indefinite functionals developed by Szulkin and Weth.

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Mathematics Subject Classification: Primary: 37J45; Secondary: 35B38, 70H05.

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