A quadratic Bolza-type problem in a non-complete Riemannian manifold

Let the nonlinear equation $D_s\dot x + \lambda \nabla_x V(x,s) = 0$ be defined in a non-complete Riemannian manifold $\m$ and consider those ones of its solutions which join any couple of fixed points in $\m$ in a fixed arrival time $T > 0$. If $V$ has a quadratic growth with respect to $x$ and if $\m$ has a convex boundary, then a ``best constant'' $\bar\lambda(T) > 0$ exists such that if $0 \le\lambda < \bar\lambda(T)$ the problem admits at least one solution while infinitely many ones exist if the topology of $\m$ is not trivial.