## 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)

### MSC:

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