Detecting multiplicity for systems of second order equations: an alternative approach

We are concerned with a system of second order differential equations of the form $x''+A(t,x)x=0, t \in [0,\pi], x \in \R^N$, where $A(t,x)$ is a symmetric $N\times N$ matrix. We focus on an asymptotically linear situation and we prove the existence of multiple solutions to the Dirichlet problem associated to the system. Multiplicity is obtained by a comparison between the number of moments of verticslity of the matrices $A_0(t)$ and $A_{\infty}(t)$ which are the uniform limits of $A(t,x)$ for $x \to 0$ and $|x| \to +\infty$, respectively. For the proof, which is based on a generalized shooting approach, we provide a theorem on the existence of zeros of a class of $N$-dimensional vector fields.