Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems (Q964988)

The author gives some parabolic-like conditions which allow further computations of the critical groups both at degenerate critical points and at infinity, and the results generalize the well-known angle conditions for the computations of the critical groups. As an application, the author considers the second-order Hamiltonian system \[ u''(t)+\nabla H(t,u(t))=0,\quad t\in \mathbb R,\tag{1} \] where \(H:\mathbb R\times \mathbb R^N\longrightarrow \mathbb R\) is \(T\)-periodic in its first variable and \(H(t,x)\) is continuous in \(t\) for each \(x\in \mathbb R^N\) and twice continuously differentiable in \(x\) for each \(t\in [0,T]\). The authors prove the existence of \(T\)-periodic solutions for the system when \(\nabla H\) is asymptotically linear both at the origin and at infinity, that is, \[ |\nabla H(t,x)-A_\infty(t)x|=0(|x|),\text{ as }|x|\rightarrow \infty,\text{ uniformly in }t\in[0,T], \] \[ |\nabla H(t,x)-A_0(t)x|=0(|x|),\text{ as }|x|\rightarrow 0,\text{ uniformly in }t\in[0,T]. \] Let \[ H_0(t,x)=H(t,x)-\tfrac{1}{2}(A_0(t)x,x), \quad H_\infty(t,x)=H(t,x)-\tfrac{1}{2}(A_\infty(t)x,x), \] where \(A_\infty(t)\) and \(A_0(t)\) are \(N\times N\) continuous symmetric matrices and \(T\)-periodic in \(t\). The main conditions are the following: {\parindent13mm \begin{itemize}\item[{\((H_{\infty})\)}] There exists \(\alpha\in (0,1]\) such that \(\lim_{|x|\rightarrow \infty} \frac{|\nabla H_{\infty}(t,x)|}{|x|^{\alpha}}=0\) uniformly for \(t\in [0,T];\) \item[{\((H_{\infty}1)\)}] \(\limsup_{|x|\rightarrow {\infty}}\frac{(\nabla H_{\infty}(t,x),x )}{|x|^{2\alpha}}\leq h_{\infty}(t)\leq 0\) uniformly for \(t\in[0,T]\), where \(h_{\infty}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{\infty}(t)\,dt<0\); \item[{\((H_{\infty}2)\)}] \(\liminf_{|x|\rightarrow {\infty}}\frac{(\nabla H_{\infty}(t,x),x )}{|x|^{2\alpha}}\geq h_{\infty}(t)\geq 0\) uniformly for \(t\in[0,T]\), where \(h_{\infty}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{\infty}(t)\,dt>0\); \item[{\((H_0)\)}] there exists \(\beta \in [1,+\infty)\) such that \(\lim_{|x|\rightarrow 0} \frac{|\nabla H_0(t,x)|}{|x|^{\beta}}=0\) uniformly for \(t\in [0,T]\); \item[{\((H_0 1)\)}] \(\limsup_{|x|\rightarrow 0}\frac{(\nabla H_{0}(t,x),x )}{|x|^{2\beta}}\leq h_0(t)\leq 0\) uniformly for \(t\in[0,T]\), where \(h_{0}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{0}(t)\,dt<0\); \item[{\((H_0 2)\)}] \(\limsup_{|x|\rightarrow 0}\frac{(\nabla H_{0}(t,x),x )}{|x|^{2\beta}}\geq h_0(t)\geq 0\) uniformly for \(t\in[0,T]\), where \(h_{0}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{0}(t)\,dt>0\); \end{itemize}} One of the main results is Theorem 1: Suppose that (\(H_\infty\)) and (\(H_0\)) hold. Assume that one of the following conditions holds: (a)\ \ \((H_\infty 1)\), \((H_0 1)\) with \(\mu_{\infty}\neq\mu_0\), (b)\ \ \((H_\infty 1)\), \((H_0 2)\) with \(\mu_{\infty}\neq\mu_0+\nu_0\), (c)\ \ \((H_\infty 2)\), \((H_0 1)\) with \(\mu_\infty +\nu_\infty\neq \mu_0 \), (d)\ \ \((H_\infty 2)\), \((H_0 2)\) with \(\mu_\infty +\nu_\infty\neq \mu_0+\nu_0 \), Then problem (1) has at least one nontrivial \(T\)-periodic solution.