Multiple periodic solutions for Hamiltonian systems with not coercive potential

Abstract

Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many periodic solutions for a class of second order Hamiltonian systems is established. Moreover, the existence of two non-trivial periodic solutions for Hamiltonian systems with not coercive potential is obtained, and the existence of three periodic solutions for Hamiltonian systems with coercive potential is pointed out. The approach is based on critical point theorems.

Ground state solutions for non-autonomous dynamical systems

“…Antonacci [3,4] gave conditions for existence of solutions with stronger constraints on the potential but without the homogeneity condition, and without the negative definite condition on the matrix. Generalizations of the above results are given by Antonacci and Magrone [2], Barletta and Livrea [6], Guo and Xu [16], Li and Zou [24], Faraci and Livrea [15], Bonanno and Livrea [7,8], Jiang [21,22], Shilgba [39,40], Faraci and Iannizzotto [14] and Tang and Xiao [53]. Some authors considered the second order system (1) where the potential function V (t, x) is quadratically bounded as |x| → ∞.…”

“…The first to consider it for potentials satisfying (3) were Berger and the author [5] in 1977. We proved the existence of solutions to (8) under the condition that V (t, x) → ∞ as |x| → ∞ uniformly for a.e. t ∈ I.…”

Homoclinic solutions of nonlinear second-order Hamiltonian systems

“…Antonacci [3,4] gave conditions for existence of solutions with stronger constraints on the potential but without the homogeneity condition, and without the negative definite condition on the matrix. Generalizations of the above results are given by Antonacci and Magrone [2], Barletta and Livrea [6], Guo and Xu [16], Li and Zou [24], Faraci and Livrea [15], Bonanno and Livrea [7,8], Jiang [21,22], Shilgba [40,41], Faraci and Iannizzotto [14] and Tang and Xiao [54]. Some authors considered the second order system (1) where the potential function V (t, x) is quadratically bounded as |x| → ∞.…”

Periodic second order superlinear Hamiltonian systems

“…Antonacci [3,4] gave conditions for existence of solutions with stronger constraints on the potential but without the homogeneity condition, and without the negative definite condition on the matrix. Generalizations of the above results are given by Antonacci and Magrone [2], Barletta and Livrea [6], Guo and Xu [13], Li and Zou [18], Faraci and Livrea [12], Bonanno and Livrea [7,8], Jiang [16,17], Shilgba [29,30], Faraci and Iannizzotto [11] and Tang and Xiao [39]. Some authors considered the second order system (1) where the potential function V (t, x) is quadratically bounded as |x| → ∞.…”