Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0840.34044
Ding, Yanheng
Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. (English)
[J] Nonlinear Anal., Theory Methods Appl. 25, No.11, 1095-1113 (1995). ISSN 0362-546X

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 $|t|\to \infty$.\par 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 $t$.\par 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.
[E.Ershov (St.Peterburg)]

MSC 2000:: *34C37 Homoclinic and heteroclinic solutions of ODE

Keywords: existence; multiplicity; homoclinic solutions; Hamiltonian systems

Cited in: Zbl pre06069271

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]