Non-minimizing connecting orbits for multi-well systems

Given a nonnegative, smooth potential $V: \mathbb{R}^k \to \mathbb{R}$ ($k \geq 2$) with multiple zeros, we say that a curve $\mathfrak{q}: \mathbb{R} \to \mathbb{R}^k$ is a connecting orbit if it solves the autonomous system of ordinary differential equations \begin{equation} \mathfrak{q}''= \nabla_{\mathbf{u}} V(\mathfrak{q}) , \hspace{2mm} \mbox{ in } \mathbb{R} \end{equation} and tends to a zero of $V$ at $\pm \infty$. Broadly, our goal is to study the existence of connecting orbits for the problem above using variational methods. Despite the rich previous literature concerning the existence of connecting orbits for other types of second order systems, to our knowledge only connecting orbits which minimize the associated energy functional in a suitable function space were proven to exist for autonomous multi-well potentials. The contribution of this paper is to provide, for a class of such potentials, some existence results regarding non-minimizing connecting orbits. Our results are closely related to the ones in the same spirit obtained by J. Bisgard in his PhD thesis (University of Wisconsin-Madison, 2005), where non-autonomous periodic multi-well potentials (ultimately excluding autonomous potentials) are considered. Our approach is based on several refined versions of the classical Mountain Pass Lemma.