Homoclinic solutions for a class of second-order singular Hamiltonian systems

Abstract

We consider a class of singular second-order Hamiltonian systems in $R^N(N\ge 2)$$\ddot{q}+\nabla V\left(q\right)=0,q\left(t\right)\notin D,$ where $V:R^N\setminus D\to R$ has a strict global maximum 0 at the origin and D ⊂  $R^N\setminus \left\{0\right\}$ is a set of singularities, that is, $V\left(q\right)\to -\infty$ as $\mathrm{dist}(q,D)\to 0$. Under the condition that D is a compact set with $C^2$-boundary and $V\left(q\right)\sim -{\left[\mathrm{dist}\right(q,D\left)\right]}^{-\alpha }$ as $\mathrm{dist}(q,D)\to 0$ for some $\alpha >0$, we show the existence of a nontrivial homoclinic solution at 0 via a suitable approximation method.