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.
Electronic Journal of Differential Equations
Texas State University, San Marcos, Department of Mathematics
Scholarly Commons Citation
Spradlin, G. S. (2004). Existence of Solutions to a Hamiltonian System Without Convexity Condition on the Nonlinearity. Electronic Journal of Differential Equations, 2004(21). Retrieved from https://commons.erau.edu/publication/245