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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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