A prescribed-energy problem for a conservative singular Hamiltonian system
This paper is concerned with the prescribed-energy problem for a singular Hamiltonian system. More precisely, for a given \(H \in \mathbb{R}\), the author proves the existence of at least one noncollision periodic solution of the equation \(\ddot q + \Delta V(q) = 0\), with \({1 \over 2} | \dot q |^ 2 + V(q) = H\), where \(q = (q_ 1, \dots, q_ N) \in \mathbb{R}^ N\) \((N \geq 3)\), and \(V(q)\) is a potential with finitely many singularities \(y_ 1, \dots, y_ d \in \mathbb{R}^ N\), satisfying the following conditions: 1. \(V(x) \in C^ 2 (\mathbb{R}^ N \backslash \{y_ 1, \dots, y_ d\}, \mathbb{R})\) is of the form \[ V(x) = \sum^ d_{i = 1} - {\alpha_ i \over | x - y_ i |^{\alpha_ i}} + W(x), \] where \(W \in C^ 2 (\mathbb{R}^ N \backslash \{y_ 1, \dots, y_ d\}, \mathbb{R})\) satisfies \(\lim_{x \to y_ i} \{| x - y_ i |^{\alpha_ i + k} \nabla^ k W(x)\} = 0\), \((k = 0,1,2)\), for all \(i = 1, \dots, d\); 2. \(\Omega = \{x \in \mathbb{R}^ N \backslash \{y_ 1, \dots, y_ d\} | V(x) < H\} \cup \{y_ 1, \dots, y_ d\}\) is bounded; 3. \(\nabla V(x) \neq 0\), for all \(x \in \partial \Omega\); 4. \(\alpha_ i \in (1,2)\) for \(i = 1, \dots, d\), if \(N \geq 4\), \(\alpha_ i \in ({4 \over 3}, 2)\) for \(i = 1, \dots, d\), if \(N = 3\).
scientific article; zbMATH DE number 754026
scientific article