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Article
Existence of Solutions to a Hamiltonian System Without Convexity Condition on the Nonlinearity
Electronic Journal of Differential Equations
(2004)
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.
Keywords
- Mountain Pass Theorem,
- variational methods,
- Nehari manifold,
- homoclinic solutions
Publication Date
2004
Citation Information
Gregory S. Spradlin. "Existence of Solutions to a Hamiltonian System Without Convexity Condition on the Nonlinearity" Electronic Journal of Differential Equations Vol. 2004 Iss. 21 (2004) p. 1 - 13 ISSN: 1072-6691 Available at: http://works.bepress.com/greg_spradlin/9/