Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems (Q961056)

The paper is concerned with the existence of homoclinic orbits of the second order Hamiltonian system \[ -\ddot{u}+L(t) u=\nabla_u R(t,u) \tag{HS} \] in \(\mathbb{R}^N\). Here \(L(t)\in\mathbb{R}^{N\times N}\) is symmetric and continuous in \(t\), and \(R:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}\) is \(C^1\) and satisfies \(0\leq R(t,u)=o(|u|)\) as \(u\to 0\). It is allowed that \(\inf_{t\in\mathbb{R}}\min\,\mathrm{spec}L(t)=-\infty\) so that the operator \(-\frac{d^2}{dt^2}+L(t)\) may not be bounded below, and the associated quadratic form on \(L^2(\mathbb{R},\mathbb{R}^n)\) is strongly indefinite. Assuming that \(R_u\) is asymptotically linear in \(u\), the authors present conditions on \(L\) and \(R\) such that (HS) has a homoclinic solution. If \(R\) is even in \(u\), they obtain multiple homoclinics. The proofs are based on a generalized linking theorem for strongly indefinite functionals due to \textit{Y. Ding} and the reviewer [Math. Nachr. 279, No. 12, 1267--1288 (2006; Zbl 1117.58007)].