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Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems. (English) Zbl 1207.34049

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: 13mm
\((H_{\infty})\)
There exists \(\alpha\in (0,1]\) such that \(\lim_{|x|\rightarrow \infty} \frac{|\nabla H_{\infty}(t,x)|}{|x|^{\alpha}}=0\) uniformly for \(t\in [0,T];\)
\((H_{\infty}1)\)
\(\limsup_{|x|\rightarrow {\infty}}\frac{(\nabla H_{\infty}(t,x),x )}{|x|^{2\alpha}}\leq h_{\infty}(t)\leq 0\) uniformly for \(t\in[0,T]\), where \(h_{\infty}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{\infty}(t)\,dt<0\);
\((H_{\infty}2)\)
\(\liminf_{|x|\rightarrow {\infty}}\frac{(\nabla H_{\infty}(t,x),x )}{|x|^{2\alpha}}\geq h_{\infty}(t)\geq 0\) uniformly for \(t\in[0,T]\), where \(h_{\infty}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{\infty}(t)\,dt>0\);
\((H_0)\)
there exists \(\beta \in [1,+\infty)\) such that \(\lim_{|x|\rightarrow 0} \frac{|\nabla H_0(t,x)|}{|x|^{\beta}}=0\) uniformly for \(t\in [0,T]\);
\((H_0 1)\)
\(\limsup_{|x|\rightarrow 0}\frac{(\nabla H_{0}(t,x),x )}{|x|^{2\beta}}\leq h_0(t)\leq 0\) uniformly for \(t\in[0,T]\), where \(h_{0}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{0}(t)\,dt<0\);
\((H_0 2)\)
\(\limsup_{|x|\rightarrow 0}\frac{(\nabla H_{0}(t,x),x )}{|x|^{2\beta}}\geq h_0(t)\geq 0\) uniformly for \(t\in[0,T]\), where \(h_{0}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{0}(t)\,dt>0\);
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.

MSC:

34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
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References:

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