Existence of homoclinic solutions for Hamiltonian systems

Abstract

We consider the Hamiltonian system $$ Ju'(x)+Mu(x)-\nabla_u F(x,u(x))=\lambda u(x), $$ where $u:\mathbb R\to \mathbb R^{2N}$. Using variational methods, we obtain existence results for homoclinic solutions by imposing conditions on $F$. These conditions are in general weaker than in the former contributions on this subject. In particular, the behaviour of $F$ with respect to $u$ may be different according to whether $u$ is small or large. The condition that $F(x,u)\not=0$ whenever $u\not=0$ is replaced by a much weaker one. In addition to the periodic case, we treat the case when $F(x,u)$ is homoclinic in $x$. Finally, the continuity of $F$ is replaced by a Carathéodory condition.