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

Wiesław Krawcewicz, Yanli Lv, Huafeng Xiao

DOI: http://dx.doi.org/10.12775/TMNA.2016.024

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 $0\lt \theta \nabla f(x)\bullet x\le \nabla^2 f(x)x\bullet x$ (for $\theta\gt 1$ 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 $p\gt 0$). As an example, we prove the existence of arbitrarily many periodic solutions with minimal period $p$ for a specific $D_n$-symmetric Newtonian system.

Keywords


Periodic solutions; Nehari manifold; Palais-Smale condition; H-fixed-point reduction

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