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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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