Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms
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.
scientific article; zbMATH DE number 5962581