Infinitely many homoclinic orbits for a class of superquadratic Hamiltonian systems

TL;DR

This paper proves the existence of infinitely many homoclinic orbits in certain superquadratic Hamiltonian systems using minimax critical point methods, even without the global Ambrosetti-Rabinowitz condition.

Contribution

It establishes the existence of infinitely many homoclinic orbits for a class of superquadratic Hamiltonian systems without requiring the global Ambrosetti-Rabinowitz condition.

Findings

Infinitely many homoclinic orbits exist for the studied systems.

Uses minimax methods in critical point theory.

Does not require the global Ambrosetti-Rabinowitz condition.

Abstract

In this paper, we prove the existence of infinitely many homoclinic orbits for the first order Hamiltonian systems $J\dot{x}-M(t)x+ R'(t,x)=0$, by the minimax methods in critical point theory, when $R(t,y)$ satisfies the superquadratic condition ${{R(t,x)}\over{\|x\r|^{2}}}\longrightarrow \pm\infty$ as $\|x\|\longrightarrow\infty$, uniformly in $t$, and need not satisfy the global Ambrosetti-Rabinowitz condition