On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

Abstract

We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in \mathbb{R}$, $q\in {\mathbb{R}}^n$, $a:\mathbb{R}\to \mathbb{R}$ is a continuous positive bounded function, $G:{\mathbb{R}}^n\to \mathbb{R}$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f:\mathbb{R}\to {\mathbb{R}}^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

15 pages, 9 figures