Multiple solutions with prescribed minimal period for second order odd Newtonian systems with symmetries

Abstract

For an orthogonal $\Gamma$-representation $V$ ($\Gamma$ is a finite group) and for an even $\Gamma$-invariant $C^2$-functional $f\colon V\to \mathbb R$ satisfying the condition $0< \theta \nabla f(x)\bullet x \leq \nabla^2 f(x)x\bullet x$ (for $\theta> 1$ and $x\in V\setminus \{0\}$), we consider the odd Newtonian system $\ddot x(t)=-\nabla f(x(t))$ and establish the existence of multiple periodic solutions with a minimal period $p$ (for any given $p > 0$). As an example, we prove the existence of arbitrarily many periodic solutions with minimal period $p$ for a specific $D_n$-symmetric Newtonian system.