Homoclinic orbits for a quasi-linear Hamiltonian system (Q1900376)

\textit{P. Rabinowitz} [Proc. R. Soc. Edinb., Sect. A 114, 33-38 (1990; Zbl 0705.34054)] obtained conditions for the existence of a homoclinic orbit through 0 of the Hamiltonian system \(u'' + V_u'(t,u) = 0\) where \(V = 1/2 < L(t)u\), \(u > + W(t,u)\) is periodic in \(t\). The paper studies an extension of this result to a quasilinear Hamiltonian system of the form \((a (|u' |^p) |u' |^{p - 2} u')' - b(|u |^p) |u |^{p - 2} u + W_u'(t,u) = 0\). The main result describes conditions for the functions \(a,b\) and \(W\) which guarantee the existence of a nontrivial homoclinic orbit. An example is given.