Existence of solutions to a Hamiltonian system without convexity condition on the nonlinearity
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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.
CitationSpradlin, G. S. (2004). Existence of solutions to a Hamiltonian system without convexity condition on the nonlinearity. Electronic Journal of Differential Equations, 2004(21), pp. 1-13.
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