Homoclinics for an almost periodically forced singular Hamiltonian system
The author considers the Hamiltonian system (HS) \(\ddot q+a(t)W'(q)=0\), where \(a\) and \(W\) satisfy \((a_1)\) \(a(t)\) is a continuous almost periodic function of \(t\) with \(a(t)\geq a_0>0\) for all \(t\in\mathbb{R}\); \((\text{W}_1)\) there is a \(\xi\in\mathbb{R}^2 \setminus\{0\}\) such that \(W\in C^2(\mathbb{R}^2\setminus \{\xi\},\mathbb{R})\); \((\text{W}_2)\) \(\lim_{x\to\xi} W(x)=-\infty\); \((\text{W}_3)\) there is a neighbourhood \({\mathcal N}\) of \(\xi\) and \(U\in C^1({\mathcal N}\setminus \{\xi\},\mathbb{R})\) such that \(|U(x)|\to\infty\) as \(x\to\xi\) and \(|U'(x)|^2\leq -W(x)\) for \(x\in{\mathcal N}\setminus \{\xi\}\); \((\text{W}_4)\) \(W(x)< W(0)=0\) if \(x\neq 0\) and \(W''(0)\) is negative definite; \((\text{W}_5)\) there is a constant \(W_0<0\) such that \(\varlimsup_{x\to\infty} W(x)\leq W_0\). A variational problem is formulated for \(E=W^{1,2}(\mathbb{R},\mathbb{R}^2)\), \(\Lambda=\{q\in E: q(t)\neq\xi\) for all \(t\in\mathbb{R}\}\) and \(I(q)=\int_\mathbb{R} (1/2\cdot |q(t)|^2- a(t)W(q))dt\), the critical points of \(I\) on \(\Lambda\) being solutions of (HS) homoclinic to 0. First the author proves that, the above conditions being fulfilled, for a given \(q\in\Lambda\) with nonzero Browder degree with respect to \(\xi\) there exists a homoclinic solution \(Q\in\Lambda\) of (HS) with \(I(Q)\in (0,I(q)]\). Then it is shown that the system (HS) has infinitely many homoclinic solutions.
scientific article; zbMATH DE number 906855