Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

Dong-Lun  Wu; Chun-Lei Tang; Xing-Ping Wu

doi:10.3934/cpaa.2016.15.57

Article Contents

2016, Volume 15, Issue 1: 57-72. Doi: 10.3934/cpaa.2016.15.57

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Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715

Received: December 2014

Revised: October 2015

Published: January 2016

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Abstract

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.

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Mathematics Subject Classification: Primary: 34C37, 37J45; Secondary: 47J30.

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