Existence of solutions to a Hamiltonian system without convexity condition on the nonlinearity
Date
2004-02-12
Authors
Spradlin, Gregory S.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We study a Hamiltonian system that has a superquadratic potential and is asymptotic to an autonomous system. In particular, we show the existence of a nontrivial solution homoclinic to zero. Many results of this type rely on a convexity condition on the nonlinearity, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. This paper replaces that condition with a different condition, which is automatically satisfied when the autonomous system is radially symmetric. Our proof employs variational and mountain-pass arguments. In some similar results requiring the convexity condition, solutions inhabit a submanifold homeomorphic to the unit sphere in the appropriate Hilbert space of functions. An important part of the proof here is the construction of a similar manifold, using only the mountain-pass geometry of the energy functional.
Description
Keywords
Mountain Pass Theorem, Variational methods, Nehari manifold, Homoclinic solutions
Citation
Spradlin, G. S. (2004). Existence of solutions to a Hamiltonian system without convexity condition on the nonlinearity. <i>Electronic Journal of Differential Equations, 2004</i>(21), pp. 1-13.
Rights
Attribution 4.0 International