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Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms. (English) Zbl 1237.37042
This paper is concerned with the multiplicity of $$2\pi$$-periodic solutions of the second order Hamiltonian system $-\ddot{x}-A(t)x=\lambda x+V'_x(t,x),$ where $$\lambda\in \mathbb R$$, $$A(t)$$ is a continuous, $$2\pi$$-periodic, symmetric matrix-valued function, and the potential $$V(t,x)$$ is $$C^2$$ with $$2\pi$$-periodicity in $$t$$, such that $V(t,0)=V'_x(t,0)=V''_x(t,0)=0$ and $0<\theta V(t,x)\leq (V'_x(t,x),x)$ for all $$t\in [0,2\pi]$$ and $$|x|$$ sufficiently large. The authors separately impose the following additional hypotheses:
(i)
$$V''_x(t,x)>0$$ for $$|x|>0$$ small and $$t\in [0,2\pi]$$.
(ii)
$$V''_x(t,x)<0$$ for $$|x|>0$$ small and $$t\in [0,2\pi]$$.
(iii)
$$V(t,x)\leq 0$$ for $$|x|>0$$ small and $$t\in [0,2\pi]$$.
It is shown, under the assumptions (i), (ii) and (iii), that for any fixed positive integer $$k$$ there is some $$\delta>0$$ such that if $\sup_{(t,x)\in [0,2\pi]\times \mathbb R^N}V^-(t,x)\leq \delta,$ then the following properties respectively hold true:
(a)
if $$\lambda\in (\lambda_k-\delta,\lambda_k)$$, there exist at least three distinct nontrivial $$2\pi$$-periodic solutions;
(b)
if $$\lambda\in (\lambda_k,\lambda_k+\delta)$$, the above problem has at least three distinct nontrivial solutions;
(c)
if $$\lambda\in (\lambda_k-\delta,\lambda_k]$$, the above problem has at least two nontrivial solutions.
The proofs depend on a careful analysis of critical groups, and the solutions are constructed by a combination of bifurcation arguments, topological linking and Morse theory.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34C25 Periodic solutions to ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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##### References:
  Bartsch, T.; Li, S.J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 419-441, (1997) · Zbl 0872.58018  Chang, K.C., Infinite dimensional Morse theory and multiple solution problems, Progr. nonlinear differential equations appl., vol. 6, (1993)  Gromoll, D.; Meyer, W., On differential functions with isolated point, Topology, 8, 361-369, (1969) · Zbl 0212.28903  Li, S.J., Periodic solutions of non-autonomous second order systems with superlinear terms, Differential integral equations, 5, 1419-1424, (1992) · Zbl 0757.34021  Li, S.J.; Liu, J.Q., Some existence theorems on multiple critical points and their applications, Kexue tongbao (Chinese), 17, 1025-1027, (1984)  Li, S.J.; Su, J.B., Existence of multiple critical points for an asymptotically quadratic functional with applications, Abstr. appl. anal., 1, 277-289, (1996) · Zbl 0938.58013  Liu, J.Q., A Morse index for a saddle point, J. systems sci. math. sci., 2, 32-39, (1989) · Zbl 0732.58011  Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York, Berlin, Heidelberg · Zbl 0676.58017  Rabinowitz, P.H., Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. sc. norm. super. Pisa cl. sci., 4, 5, 215-223, (1978) · Zbl 0375.35026  Rabinowitz, P.H., Periodic solutions of Hamiltonian systems, Comm. pure appl. math., 31, 157-184, (1978) · Zbl 0358.70014  Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS reg. conf. ser. math., vol. 65, (1986), AMS Providence, RI · Zbl 0609.58002  Rabinowitz, P.H.; Su, J.B.; Wang, Z.-Q., Multiple solutions of superlinear elliptic equations, Rend. lincei mat. appl., 18, 97-108, (2007) · Zbl 1223.35173  Su, J.B., Multiplicity results for asymptotically linear elliptic problems at resonance, J. math. anal. appl., 278, 397-408, (2003) · Zbl 1290.35109  Su, J.B., Nontrivial critical points for asymptotically quadratic functional at resonance, (), 225-234 · Zbl 1048.58009  Su, J.B.; Zhao, L.G., Multiple periodic solutions of ordinary differential equations with double resonance, Nonlinear anal., 70, 1520-1527, (2009) · Zbl 1219.34056  Wang, Z.-Q., On a superlinear elliptic equation, Ann. inst. H. Poincaré anal. non lineaire, 8, 43-57, (1991) · Zbl 0733.35043
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