Topological Methods in Nonlinear Analysis

Some existence results for dynamical systems on non-complete Riemannian manifolds

Elvira Mirenghi and Maria Tucci

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Let $\mathcal M^*$ be a non-complete Riemannian manifold with bounded topological boundary and $V\colon \mathcal M \to \mathbb R$ a $C^2$ potential function subquadratic at infinity.

In this paper we look for curves $x\colon [0,T]\to\mathcal M$ having prescribed period $T$ or joining two fixed points of $\mathcal M$, satisfying the system $$ D_t (\dot x(t))=-\nabla_R V(x(t)), $$ where $D_t(\dot x(t))$ is the covariant derivative of $\dot x$ along the direction of $\dot x$ and $\nabla_R V$ the Riemannian gradient of $V$.

We assume that $V(x) \to -\infty$ if $d(x,\partial\mathcal M)\to 0$ and, in the periodic case, suitable hypotheses on the sectional curvature of $\mathcal M$ at infinity.

We use variational methods in addition with a penalization technique and Morse index estimates.

Article information

Topol. Methods Nonlinear Anal., Volume 13, Number 1 (1999), 163-180.

First available in Project Euclid: 29 September 2016

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Mirenghi, Elvira; Tucci, Maria. Some existence results for dynamical systems on non-complete Riemannian manifolds. Topol. Methods Nonlinear Anal. 13 (1999), no. 1, 163--180.

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