Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 21, pp. 1-13.

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

Author: Gregory S. Spradlin (Embry-Riddle Aeronautical Univ.,Daytona Beach, Florida, USA)

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.

Submitted October 31, 2003. Published February 12, 2004.
Math Subject Classifications: 34C37, 47J30.
Key Words: Mountain Pass Theorem; variational methods; Nehari manifold; 
           homoclinic solutions.