Relative Morse index and its application to Hamiltonian systems in the presence of symmetries
A Hamiltonian system (1) \(\dot z= JH'(z, t)\), \(z\in \mathbb{R}^{2N}\), where \(H\in C^1(\mathbb{R}^{2N}\times \mathbb{R}, \mathbb{R})\) and \(H'\) denotes the gradient of \(H\) with respect to \(z\), is considered. It is assumed that \(H'(z, t)= B_0(t)+ o(|z|)\) as \(|z|\to 0\) uniformly in \(t\), \(H'(z, t)= B_\infty(t)+ o(|z|)\) as \(|z|\to \infty\) uniformly in \(t\), \(H(z, t)= H(- z, t)\), \(H(z, t+ 1)= H(z, t)\), \(\forall z\in \mathbb{R}^{2N}\), where \(B_0(t)\) and \(B_\infty(t)\) are symmetric matrices, continuous and 1-periodic in \(t\). By \((i_0, n_0)\) and \((i_\infty, n_\infty)\) the so-called Maslov-type indices of \(B_0\) and, \(B_\infty\) respectively, are denoted. The notion of the Maslov-type index comes from a paper of Y. Long and is a modification of the concept of Maslov index introduced by Conley and Zehnder. The reviewed paper provides its definition. Using a theorem of Benci the following results are proved: Theorem: If \(n_0= n_\infty= 0\) then (1) has at least \(|i_0- i_\infty|\) pairs of nontrivial 1-periodic solutions. Theorem: If the function \(V(z, t):= H(z, t)- {1\over 2} (B_\infty(t) z, z)\) tends to \(\infty\) as \(|z|\to \infty\) uniformly in \(t\) and \(V'\) is bounded then (1) has at least \(i_0- i_\infty- n_\infty\) pairs of nontrivial 1-periodic solutions whenever \(i_0- i_\infty- n_\infty> 0\). Theorem: If \(r> 0\), \(p\in (1, 2)\), \(a_1> 0\), and \(a_2> 0\) such that \(pV(z, t)\geq (z, V'(z, t))> 0\), \(|z|\geq r\), \(t\in \mathbb{R}\), \(\varlimsup_{|z|\to \infty} |z|^{- 1}|V'(z, t)|\leq c< {1\over 2}\), \(V(z, t)\geq a_1|z|^p- a_2\), then (1) has at least \(i_0- i_\infty- n_\infty\) pairs of nontrivial 1-periodic solutions whenever \(i_0- i_\infty- n_\infty> 0\). Under additional assumption \(H(z, t)\leq H(0, t)\) for every \(z\in \mathbb{R}^{2N}\) and \(t\in \mathbb{R}\), conclusions on the existence of nonconstant solutions in the above theorems are also given.
scientific article; zbMATH DE number 823552
scientific article