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

Abstract

In this paper we are concerned with a system of second-order differential equations of the form $x''+A(t,x)x=0$, $t\in [0,\pi]$, $x\in {{\bf R}}^N$, where $A(t,x)$ is a symmetric $N\times N$ matrix. We concentrate 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 verticality 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.