an:05962581
Zbl 1237.37042
Li, Xiaoli; Su, Jiabao; Tian, Rushun
Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms
EN
J. Math. Anal. Appl. 385, No. 1, 1-11 (2012).
00287898
2012
j
37J45 34C25 58E05
Hamiltonian systems; homological linking; Morse theory; periodic solutions; bifurcation method
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: {\parindent=8mm\begin{itemize}\item[(i)] \(V''_x(t,x)>0\) for \(|x|>0\) small and \(t\in [0,2\pi]\).\item[(ii)] \(V''_x(t,x)<0\) for \(|x|>0\) small and \(t\in [0,2\pi]\).\item[(iii)] \(V(t,x)\leq 0\) for \(|x|>0\) small and \(t\in [0,2\pi]\).
\end{itemize}} 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:{\parindent=6mm\begin{itemize}\item[(a)] if \(\lambda\in (\lambda_k-\delta,\lambda_k)\), there exist at least three distinct nontrivial \(2\pi\)-periodic solutions; \item[(b)] if \(\lambda\in (\lambda_k,\lambda_k+\delta)\), the above problem has at least three distinct nontrivial solutions;\item[(c)] if \(\lambda\in (\lambda_k-\delta,\lambda_k]\), the above problem has at least two nontrivial solutions.
\end{itemize}} 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.
Chun-Lei Tang (Chongqing)