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Subharmonic oscillations of convex Hamiltonian systems. (English) Zbl 0579.34030

We prove the following results about subharmonic oscillations of subquadratic convex Hamiltonian systems by combining the dual least action principle and some estimates. Theorem 1. Let \(H(t,u)\) be \(T\)-periodic in \(t\) and convex in \(u\). If \(H(t,u)\to +\infty\), \(| u| \to +\infty\), \(H(t,u)/| u|^ 2\to 0,\) \(| u| \to +\infty\) then for every \(k\in {\mathbb{N}}^*\) there exists a kT-periodic solution \(u_ k\) of \(J\dot u+D_ uH(t,u)=0\) such that \(| U_ k|_{\infty}\to \infty\) and the minimal period of \(U_ k\) tends to infinity as \(k\to +\infty.\)
Theorem 2. Let \(V(t,u)\) be \(T\)-periodic in \(t\) and convex in \(u\). If \(V(t,u)\to +\infty\), \(| u| \to +\infty\), \(V(t,u)/| u|^ 2\to 0\), \(| u| \to +\infty\) then for every \(k\in {\mathbb{N}}^*\) there exists a kT-periodic solution \(u_ k\) of \(\ddot u+D_ uV(t,u)=0\) such that \(| u_ k|_{\infty}\to \infty\) and the minimal period of \(u_ k\) tends to infinity as \(k\to +\infty.\)
Theorem 1 generalizes a result of P. H. Rabinowitz [Comm. Pure Appl. Math. 33, 609-633 (1980; Zbl 0425.34024)] where it is assumed that i) \(| u| \geq R\Rightarrow 0<(u,H_ u(t,u))\leq V H(t,u),\) \(V\in ]1,2[\); ii) \(a_ 1| u|^ V-a_ 2\leq H(t,u);\) iii) \(H\) is strictly convex in \(u\).
Theorem 2 generalizes a result of F. H. Clarke and F. Ekeland [Arch. Rat. Mech. Anal. 78, 315-333 (1982; Zbl 0514.34032)] where it is assumed that i) V(t,u)\(\geq V(0,0)\); ii) \(a_ 1| u|^{1+\alpha}- a_ 2\leq V(t,u)\leq a_ 3| u|^{2-\gamma}+a_ 4;\) iii) \(| u| \leq \eta \Rightarrow v(t,u)\geq K| u|^ 2/2;\) iv) \(V\) is strictly convex in \(u\).

MSC:

34C25 Periodic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E30 Variational principles in infinite-dimensional spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
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[1] Brezis, H.; Coron, J. M., Periodic solutions of non linear wave equations and hamiltonian systems, Am. J. Math., 103, 559-570 (1980) · Zbl 0462.35004
[2] Brezis, H.; Nirenberg, L., Characterizations of the ranges of some non linear operators and applications to boundary value problems, Annali Scu. norm. sup. Pisa, 5, 225-325 (1978) · Zbl 0386.47035
[3] Clarke, F.; Ekeland, I., Hamiltonian trajectories having prescribed minimal period, Communs pure appl. Math., 33, 103-115 (1980) · Zbl 0403.70016
[4] Clarke, F.; Ekeland, I., Nonlinear oscillations and boundary value problems for hamiltonian systems, Archs ration. Mech. Analysis, 78, 315-333 (1982) · Zbl 0514.34032
[5] Dolph, C. L., Nonlinear integral equations of the Hammerstein type, Trans. Am. math. Soc., 66, 289-307 (1949) · Zbl 0036.20202
[6] Rabinowitz, P., On subharmonic solutions of hamiltonian systems, Communs pure appl. Math., 33, 609-633 (1980) · Zbl 0425.34024
[7] Willem, M., Remarks on the dual least action principle, Z. Analysis Anwend., 1, 85-90 (1982) · Zbl 0528.58006
[8] Willem, M., Periodic oscillations of odd second order hamiltonian systems, Boll. Un. Mat. Ital., 3B, 233-304 (1984) · Zbl 0582.58014
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