# New homoclinic orbits for Hamiltonian systems with asymptotically quadratic growth at infinity

## TL;DR

This paper establishes the existence and multiplicity of homoclinic solutions in Hamiltonian systems with asymptotically quadratic nonlinearities, introducing new conditions and embedding theorems to extend prior results.

## Contribution

It introduces a new coercive condition and embedding theorem to prove the existence of multiple homoclinic orbits in such Hamiltonian systems.

## Key findings

- At least one nontrivial homoclinic orbit exists.
- Infinitely many homoclinic orbits are found under symmetry conditions.
- New asymptotically quadratic conditions differ from previous studies.

## Abstract

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.