HOMOCLINIC SOLUTIONS FOR SUBQUADRATIC HAMILTONIAN SYSTEMS WITHOUT COERCIVE CONDITIONS

In this paper we investigate the existence and multiplicity of classical homoclinic solutions for the following second order Hamiltonian systems $$ (\mbox{HS}) \ddot u-L(t)u+\nabla W(t,u)=0, $$ where $L\in C(\mathbb R,\mathbb R^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb R$, $W\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$ and $\nabla W(t,u)$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming that $L$ is bounded in the sense that there are constants $0<\tau_1<\tau_2$ such that $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ for all $(t,u)\in \mathbb R\times \mathbb R^n$ and $W(t,u)$ is of subquadratic growth at infinity, we are able to establish two new criteria to guarantee the existence and multiplicity of classical homoclinic solutions for (HS), respectively. Recent results in the literature are extended and significantly improved.