Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms.

*(English)*Zbl 1237.37042This 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 \(t\in [0,2\pi]\).

- (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.

Reviewer: Chun-Lei Tang (Chongqing)

##### 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 |

##### Keywords:

Hamiltonian systems; homological linking; Morse theory; periodic solutions; bifurcation method
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\textit{X. Li} et al., J. Math. Anal. Appl. 385, No. 1, 1--11 (2012; Zbl 1237.37042)

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