## 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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