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Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. (English) Zbl 0840.34044
The existence of solutions homoclinic to an equilibrium $$0\in \mathbb{R}^N$$ for second-order time-dependent Hamiltonian systems of the type $\ddot q- L(t) q+ {\partial\over \partial q} W(t, q)= 0,\quad q\in \mathbb{R}^N,\tag{1}$ where $$L\in C(\mathbb{R}, \mathbb{R}^{N^2})$$ is a symmetric matrix, $$W\in C^1(\mathbb{R}\times \mathbb{R}^N, \mathbb{R})$$, is studied. As usual, a solution $$q(t)$$ of system (1) is said to be homoclinic (to 0) if $$q(t)\neq 0$$, $$q(t)\to 0$$, and $$\dot q(t)\to 0$$ as $$|t|\to \infty$$.
The existence and multiplicity of homoclinic solutions for Hamiltonian systems of this type have been studied in many recent papers via the critical point theory under the assumptions that $$L(t)$$ is positive definite for all $$t\in \mathbb{R}$$, $$W(t, q)$$ is globally superquadratic in $$q$$ and the potential $$V(t, q)= -{1\over 2} L(t)q\cdot q+ W(t, q)$$ is periodic in $$t$$.
The author studies the existence of homoclinic (to 0) solutions for a class of systems (1) when the global positive definiteness of $$L(t)$$ is not necessarily satisfied and $$L$$ and $$W$$ are not periodic in $$t$$. Both the case that $$W(t, q)$$ is superquadratic in $$q$$ and the one that $$W(t, q)$$ is of subquadratic growth as $$|q|\to \infty$$ are considered.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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