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A multibump construction in a degenerate setting. (English) Zbl 0876.34055

The author studies the existence of multibump homoclinic solutions of the singular Hamiltonian system (HS) \(\ddot{q}+V_q(t,q)=0\) on \(\mathbb{R} ^2\). Here \(V\) is defined on \(\mathbb{R} \times (\mathbb{R}^2\setminus \{\xi\})\) with \(\xi\neq 0\) the singularity and \(V(t,q)<V(t,0)\equiv 0\) for \(q\neq 0\). It is assumed that \(V\) satisfies the strong force condition, is bounded away from \(0\) for \(|q|\to \infty \) and \(V_{qq}(t,0)\) is negative definite for any \(t \in \mathbb{R}\). Under these conditions homoclinic orbits of (HS) correspond to critical points of a functional \(I\), and a minimization argument yields a homoclinic solution \(Q\) that winds positively around \(\xi \). In an earlier paper [Chin. J. Math. 24, No. 1, 1-36 (1996)] the author proved the existence of multibump solutions of (HS) that are close to sums of translates of \(Q\) provided \(Q\) is an isolated critical point of \(I\). This last condition is now considerably weakened. Let \({\mathcal{S}}\) denote the connected component of \(Q\) in the set of nontrivial critical points of \(I\). The main results states that for the existence of the multibump solutions it essentially suffices that the set \(\{q(0) : q \in {\mathcal{S}} \} \subset \mathbb{R}^2\) is bounded away from \(0\). This condition excludes the case of autonomous systems.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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