MULTIPLICITY OF HOMOCLINIC ORBITS IN QUASI-LINEAR AUTONOMOUS LAGRANGIAN SYSTEMS

作者：Buffoni Boris^*; Landry Laurent

来源：Discrete and Continuous Dynamical Systems, 2010, 27(1): 75-116.

DOI：10.3934/dcds.2010.27.75

• Homoclinic orbits
• Lagrangian systems
• variational methods
• critical-point theory
• minimax method

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
q + V' (q) = 0, q is an element of C(2) (R, R(m)), m >= 2
where V : R(m) -> R is a function of the form
V(q) = -vertical bar q vertical bar(2)/2 +W(q)
with W is an element of 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, (q) over dot) that weakly depend on (q) over dot. In this case an homoclinic orbit corresponds to a classical solution to the equation
q - q + W(1)(q, (q) over dot) - d/dt W2(q, (q) over dot) = 0,
where W(i) = partial derivative(i)W for i = 1, 2.

• 出版日期2010-5