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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 $0\in \bbfR^N$ for second-order time-dependent Hamiltonian systems of the type $$\ddot q- L(t) q+ {\partial\over \partial q} W(t, q)= 0,\quad q\in \bbfR^N,\tag1$$ where $L\in C(\bbfR, \bbfR^{N^2})$ is a symmetric matrix, $W\in C^1(\bbfR\times \bbfR^N, \bbfR)$, is studied. As usual, a solution $q(t)$ of system (1) is said to be homoclinic (to 0) if $q(t)\ne 0$, $q(t)\to 0$, 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 $t\in \bbfR$, $W(t, q)$ 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.

34C37Homoclinic and heteroclinic solutions of ODE
Full Text: DOI
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