# American Institute of Mathematical Sciences

January  2016, 15(1): 57-72. doi: 10.3934/cpaa.2016.15.57

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

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

Received  December 2014 Revised  October 2015 Published  December 2015

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.
Citation: Dong-Lun Wu, Chun-Lei Tang, Xing-Ping Wu. Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57
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