Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems (Q968737)

Let \(m\geq 2\) and the potential \(V:\mathbb{R}^m\rightarrow \mathbb{R}, V(q)=-|q|^2/2+W(q, \dot{q})\). A set of five hypothesis about \(W\) are considered in order to prove the existence of at least two non-trivial homoclinic orbits for the Lagrangian system associated to \(V\). The autonomous case \(W=W(q)\) covers the results of \textit{A. Ambrosetti} and \textit{V. Coti Zelati} [Rend. Semin. Mat. Univ. Padova 89, 177--194 (1993; Zbl 0806.58018)] where \(W\in C^2(\mathbb{R}^m, \mathbb{R})\) is superquadratic, satisfies a ``pinching'' hypothesis and also an hypothesis on its second derivative.