Electron. J. Diff. Eqns., Vol. 2004(2004), No. 21, pp. 1-13.

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

Gregory S. Spradlin

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.

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
Embry-Riddle Aeronautical University
Daytona Beach, Florida 32114-3900, USA
e-mail: spradlig@erau.edu

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