# American Institute of Mathematical Sciences

2003, 2003(Special): 173-181. doi: 10.3934/proc.2003.2003.173

## A quadratic Bolza-type problem in a non-complete Riemannian manifold

 1 Dipartimento Di Matematica, Universita' degli Studi di Bari "Aldo Moro", via E. Orabona 4, 70125 Bari, Italy 2 Departamento de Álgebra, Geometría y Topología, Facultdad de Ciencias, Universidad de Granada, Avenida Fuentenueva s/n, 18071 Granada, Spain, Spain

Received  September 2002 Revised  March 2003 Published  April 2003

Let the nonlinear equation $D_s(dotx) + \lambda \nabla_x V (x, s) = 0$ be defined in a non–complete Riemannian manifold $M$ and consider those ones of its solutions which join any couple of fixed points in $M$ in a fixed arrival time $T > 0$. If $V$ has a quadratic growth with respect to $x$ and if $M$ has a convex boundary, then a "best constant" $bar(\lambda)(T)>$ 0 exists such that if $0 \<= \lambda \< bar(\lambda)(T)$ the problem admits at least one solution while infinitely many ones exist if the topology of $M$ is not trivial.
Citation: Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173
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