Girardi, Mario; Matzeu, Michele Periodic solutions of second order nonautonomous systems with the potentials changing sign. (English) Zbl 0799.58064 Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 4, No. 4, 273-277 (1993). This paper is concerned with the following second order system \[ \ddot x + b(t) V'(x) = 0,\quad \text{for\;}x \in \mathbb{R}^ N, \] where \(V\in C^ 2(\mathbb{R}^ N,\mathbb{R})\) satisfies some superquadratic growth assumptions and \(b\) is a \(T\)-periodic continuous real function. Some existence results for periodic solutions of the above system are presented in Theorems 1 and 2 with positive and negative average values for \(b\), respectively. The proofs are based on the well-known techniques of critical point theory. The multiplicity results are given in Theorems 3 and 4, based on the consideration of the Nehari’s manifold associated with the functional considered. Reviewer: Tongren Ding (Beijing) Cited in 4 Documents MSC: 37G99 Local and nonlocal bifurcation theory for dynamical systems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 34C25 Periodic solutions to ordinary differential equations Keywords:potentials changing sing; second order differential equations; periodic solutions PDFBibTeX XMLCite \textit{M. Girardi} and \textit{M. Matzeu}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 4, No. 4, 273--277 (1993; Zbl 0799.58064) Full Text: EuDML