Existence of solutions to a Hamiltonian system without convexity condition on the nonlinearity

Date

2004-02-12

Authors

Spradlin, Gregory S.

Journal Title

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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.

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Attribution 4.0 International

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