Homoclinic orbits of first-order superquadratic Hamiltonian systems

TL;DR

This paper investigates the existence of infinitely many large energy homoclinic orbits in first-order superquadratic Hamiltonian systems, employing fountain theorems and Nehari manifold methods under various superquadraticity conditions.

Contribution

It establishes new existence results for homoclinic orbits in Hamiltonian systems without relying on the classical Ambrosetti-Rabinowitz condition.

Findings

Existence of infinitely many large energy homoclinic orbits when H is even in u.

Existence of a ground state homoclinic orbit without the Ambrosetti-Rabinowitz condition.

Application of generalized fountain theorems and Nehari manifold techniques.

Abstract

In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system {equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}. {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 due to 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.