Multiple Homoclinics for Nonperiodic Damped Systems with Superlinear Terms

Abstract

We study the following second-order differential system

$$\begin{aligned} -\ddot{u}(t)-M\dot{u}(t)+L(t)u(t)=H_u(t,u(t)),\quad t\in \mathbb {R}, \end{aligned}$$

(0.1)

which can be regarded as a second-order Hamiltonian system with a damped term. Here, the nonlinearity H(t, u) is superquadratic as $|u|\rightarrow \infty$. We do not need any periodic conditions, and we obtain infinitely many nontrivial homoclinic orbits of this system by variational methods. Our result improves and extends the corresponding results existed.

Appendix

Here, we give an example with infinitely many nontrivial homoclinic orbits for the system (1.2) in the special case $N=2$.

For $N=2$, we let $u=u(t)=(\begin{array}{c}u_1(t)\\ u_2(t)\end{array})=(\begin{array}{c}u_1\\ u_2\end{array}), L(t)=(t^4\sin ^2t+1)(\begin{array}{cc}1\quad 0\\ 0\ \quad 1\end{array}), M=(\begin{array}{cc}0\quad -1\\ 1\ \quad 0\end{array})$ and $H_u(t,u)=\nu P(t)|u|^{\nu -2}u$ with $P(t)=\frac{b}{\nu -2}e^{|t|}\ge \frac{b}{\nu -2} (b$ and $\nu$ are the constants in Corollary 1.1), then (1.2) becomes to

$$\begin{aligned} -\left( \begin{array}{c}\ddot{u}_1\\ \ddot{u}_2\end{array}\right) -\left( \begin{array}{cc}0\quad -1\\ 1\ \quad 0\end{array}\right) \left( \begin{array}{c}\dot{u}_1\\ \dot{u}_2\end{array}\right) +(t^4\sin ^2t+1)\left( \begin{array}{c}u_1\\ u_2\end{array}\right) =\nu P(t)|u|^{\nu -2}\left( \begin{array}{c}u_1\\ u_2\end{array}\right) . \end{aligned}$$

Obviously, Theorem 1.1 and Corollary 1.1 imply the above system has infinitely many nontrivial homoclinic orbits.