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Solutions of nonperiodic super quadratic Hamiltonian systems. (English) Zbl 1346.49016

Summary: This paper concerns solutions for the Hamiltonian system: \(\dot{u}=\mathcal JH_u(t,u)\), where \(H(t,u)=1/2Lu\cdot u+W(t,u), L\) is a \(2N\times 2N\) symmetric matrix, and \(W\in C^1(\mathbb R\times\mathbb R^{2N},\mathbb R)\). We consider the case that \(0\notin \sigma _{c}( - (Jd/dt+L))\) and \(W\) satisfies some new generalized super quadratic condition different from the type of Ambrosetti-Rabinowitz. The method is variational: by virtue of some auxiliary system related to the “limit equation” of the Hamiltonian system, we first establish that the \((C)_{c}\)-condition holds true for all \(c\) less than the least energy of the limit equation. Then, using some recently developed weak linking theorem, we obtain a least energy solution of the Hamiltonian system.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
70H05 Hamilton’s equations
70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
70G75 Variational methods for problems in mechanics
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