Electron. J. Diff. Eqns., Vol. 2004(2004), No. 21, pp. 113.
Existence of solutions to a Hamiltonian system without convexity
condition on the nonlinearity
Gregory S. Spradlin
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 mountainpass 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 mountainpass geometry of the energy functional.
Submitted October 31, 2003. Published February 12, 2004.
Math Subject Classifications: 34C37, 47J30.
Key Words: Mountain Pass Theorem, variational methods, Nehari manifold,
homoclinic solutions.
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Gregory S. Spradlin
Department Mathematics
EmbryRiddle Aeronautical University
Daytona Beach, Florida 321143900, USA
email: spradlig@erau.edu 
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