Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials

Ziheng Zhang; Rong Yuan

doi:10.3934/cpaa.2014.13.623

Article Contents

2014, Volume 13, Issue 2: 623-634. Doi: 10.3934/cpaa.2014.13.623

This issue Previous Article Well-posedness of abstract distributed-order fractional diffusion equations Next Article A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities

Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials

Ziheng Zhang^1, and
Rong Yuan^2,

1.
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387
2.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: December 31, 2012

Revised: March 31, 2013

Published: February 2014

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Abstract

In this paper we investigate the existence of infinitely many homoclinic solutions for the following damped vibration problems \begin{eqnarray} \ddot q+A \dot q-L(t)q+W_q(t,q)=0, \end{eqnarray} where $A$ is an antisymmetric constant matrix, $L\in C(R,R^{n^2})$ is a symmetric and positive definite matrix for all $t\in R$, $W\in C^1(R\times R^n,R)$. The novelty of this paper is that, for the case that $W$ is subquadratic at infinity, we establish two new criteria to guarantee the existence of infinitely many homoclinic solutions for (DS) via the genus properties in critical point theory. Recent results in the literature are generalized and significantly improved.

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Mathematics Subject Classification: Primary: 34C37, 35A15; Secondary: 35B38.

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