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Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems

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  • 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


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

    Mathematics Subject Classification: Primary: 37J45, 58E50.


    \begin{equation} \\ \end{equation}
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