Spradlin, Gregory S.2021-04-052021-04-052004-02-12Spradlin, G. S. (2004). Existence of solutions to a Hamiltonian system without convexity condition on the nonlinearity. <i>Electronic Journal of Differential Equations, 2004</i>(21), pp. 1-13.1072-6691https://hdl.handle.net/10877/13340We 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.Text13 pages1 file (.pdf)enAttribution 4.0 InternationalMountain Pass TheoremVariational methodsNehari manifoldHomoclinic solutionsArticle