A Multiplicity Theorem for a Perturbed Second-Order Non-Autonomous System

Abstract

In this paper we establish a multiplicity result for a second-order non-autonomous system. Using a variational principle of Ricceri we prove that if the set of global minima of a certain function has at least k connected components, then our problem has at least k periodic solutions. Moreover, the existence of one more solution is investigated through a mountain-pass-like argument.

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| → ∞.…”

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| → ∞.…”

“…Since λ l−1 ≤ 0, these theorems allow linear, sublinear and superlinear growth at infinity for the problem (11). In particular, potentials ofthe form b(t)|x| p are included.…”

“…It will not be included in the interval λ ∈ [µ/λ l , ∞) unless µ ≤ λ l . The monotonicity trick enables one to solve problems such as (11) for almost all λ in an interval when the hypotheses are too weak to solve it for definite values of λ. By adding hypotheses, one is able to solve them for specific values of λ.…”