Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems

Pages: 75 - 116, Volume 27, Issue 1, May 2010

doi:10.3934/dcds.2010.27.75       Abstract        Full Text (378.1K)       Related Articles

Boris Buffoni - Mathematics Section (SB/IACS/CAA), Ecole Polytechnique Fédérale - Lausanne, Station 8, CH 1015 Lausanne, Switzerland (email)
Laurent Landry - Le Grand-Chemin 92, 1066 Epalinges, Switzerland (email)

Abstract: The existence of at least two homoclinic orbits is proved by A. Ambrosetti and V. Coti Zelati (Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194) for autonomous Lagrangian systems

$\ddot{q}+V'(q)=0, ~q\in C^2(\R,\R^m),~m\geq 2 $

where $V:\R^m\rightarrow\R$ is a function of the form

$ V(q)=-\frac{|q|^2}{2}+W(q) $

with $W\in C^2(\R^m,\R)$ superquadratic, satisfying a "pinching'' hypothesis and an hypothesis on its second derivative.
   The present work deals with potentials of the form $W(q,\dot{q})$ that weakly depend on $\dot{q}$. In this case an homoclinic orbit corresponds to a classical solution to the equation

$\ddot{q}-q+W_1(q,\dot{q})-\frac{d}{dt}W_2(q,\dot{q})=0,$

where $W_i=\partial_i W$ for $i=1,2$.

Keywords:  Homoclinic orbits, Lagrangian systems, variational methods, critical-point theory, minimax method.
Mathematics Subject Classification:  Primary: 37J45, 58E50.

Received: February 2009;      Revised: November 2009;      Published: February 2010.