Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

Abstract

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems $\ddot{u}+a\left(t\right)W_u\left(u\right)=0$, (HS) where , $u=\left(u_1,u_2, \dots ,u_N\right)\in {ℝ}^N\left(N\ge 3\right)$, $a:ℝ\to ℝ$ is a continuous bounded function, and the potential $W:{ℝ}^N\setminus \{\xi \}\to ℝ$ has a singularity at $0\ne \xi \in {ℝ}^N$, and $W_u\left(u\right)$ is the gradient of $W$ at $u$. The novelty of this paper is that, for the case that $N$ $\ge 3$ and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of $W$. Different from the cases that (HS) is autonomous $\left(a\left(t\right)\equiv 1\right)$ or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and $N$ $\ge 3$. Besides the usual conditions on $W$, we need the assumption that for all $t\in ℝ$ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.