On the existence of homoclinic orbits for a second-order Hamiltonian system. (English) Zbl 0894.34043

The author deals with the system \[ d^2q(t)/dt^2-a(t){| }q{| }^{p-2}q+\nabla W(t,q)\break =0, \] where \(p>2\), \(q\in \mathbb{R}^N\), and \(a(t)\) is a continuous function satisfying \[ a(t)\geq \gamma {| }t{| }^\alpha \quad \biggl(\alpha >\frac {p-2}2,\gamma >0\biggr). \] Using the crucial fact that the functional \[ f(q)=\int \Big (\frac 12{| }d q/d t{| }^2+\frac {a(t)}p{| }q{| }^p -W\Big)d t \] satisfies the Palais-Smale condition in the norm \[ \max \bigg (\Big (\int \big ({| }q{| }^2+{| }d q/d t{| }^2\big)d t\Big)^{1/2}, \Big (\int a(t){| }q{| }^pd t\Big)^{1/2}\bigg), \] he proves the existence of an infinite number of homoclinic solutions \[ \lim \big {| }q(t)\big {| }=\lim \big {| }d q(t)/d t\big {| } =0\quad ({| }t{| }\to \infty) \] in two subcases: either \(E=\frac \lambda 2{| }q{| }^2\;(\lambda >0)\), or \(0<\mu W<q\nabla W\;(\mu >p)\), \(\nabla W=o\big ({| }q{| }^{p-1}\big)\) as \({| }q{| }\to 0\) uniformly in \(t\), and \({| }W{| }+{| }\nabla W{| }\leq V\) for appropriate \(V=V(q)\) independent of \(t\).
Reviewer: J.Chrastina (Brno)


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems