Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials
Considering the equation (1) \(q''(t)+V''(q(t))=0,\) \(q\in C^ 2(R,R^ N\setminus \{0\})\), where \(V'(x)\) is the gradient of V at x and \(V(x)=- 1/| x|^{\alpha}+W,\) \(W\in C^ 1(R^ N,R)\), \(\alpha >0\), the following main result is given. Theorem. Let \(0<\rho <R\), \(h\in R\), \(V\in C^ 1(\{x\in R^ N:\rho \leq | x| \leq R\},R)\), \(N\geq 2\) such that (i) \(h-V(x)-<V'(x),x>\leq 0,\) \(\forall x:| x| =\rho,\) (ii) \(h-V(x)-<V'(x),x>\geq 0,\) \(\forall x:| x| =R,\) (iii) h-V(x)0, \(\forall x:\rho \leq | x| \leq R.\) Then there exists at least one nonconstant periodic solution q(t) of (1) having energy h and such that \(\rho\leq | q(t)| RoR\), \(\forall t\in R\).
scientific article; zbMATH DE number 4129227
scientific article