Abstract
In this paper we consider a class of second order Hamiltonian system with the nonlinearity of linear growth. Compared with the existing results, we do not assume an asymptotic of the nonlinearity at infinity to exist. Moreover, we allow the system to be resonant at zero. Under some general conditions, we will establish the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and two critical point theorems.
Highlights
Consider the following second order Hamiltonian systems − x = Vx(t, x), (1.1)where V ∈ C2(R × RN, R) with V(t + T, x) = V(t, x) for some T > 0
During the past forty years, the existence and multiplicity of periodic solutions for second order Hamiltonian systems have been extensively studied by variational methods
Liu where for two symmetric matrices A and B, A ≤ B means that B − A is semi-positively definite. Under this general linear growth condition, we will construct a sequence of approximate systems and use the Morse theory and two critical point theorems to establish the existence and multiplicity of nontrivial periodic solutions for the system
Summary
Under this general linear growth condition, we will construct a sequence of approximate systems and use the Morse theory and two critical point theorems to establish the existence and multiplicity of nontrivial periodic solutions for the system. Ik(x) ∈ C2(E, R) and the critical points of Ik correspond to the periodic solutions of the following system
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