Some existence results for dynamical systems on non-complete Riemannian manifolds

Abstract

Let $\mathcal M^*$ be a non-complete Riemannian manifold with bound-ed topological boundary and $V: \mathcal M \to \mathbb R$ a $C^2$ potential function subquadratic at infinity. In this paper we look for curves $x: [0,T]\to\mathcal M$ having prescribed period $T$ or joining two fixed points of $\mathcal M$, satisfying the system $$ D_t (\dot x(t))=-\nabla_R V(x(t)), $$ where $D_t(\dot x(t))$ is the covariant derivative of $\dot x$ along the direction of $\dot x$ and $\nabla_R V$ the Riemannian gradient of $V$. We assume that $V(x) \to -\infty$ if $d(x,\partial\mathcal M)\to 0$ and, in the periodic case, suitable hypotheses on the sectional curvature of $\mathcal M$ at infinity. We use variational methods in addition with a penalization technique and Morse index estimates.