Nontrivial solutions for superquadratic nonautonomous periodic systems
The authors consider a nonautonomous second order periodic system in \({\mathbb R}^n\) of the form \[ -x''(t)-A(t)x(t)=\nabla F(t,x(t)), \] where \(A:[0,b] \to {\mathbb R}^{n \times n}\) is continuous and such that \(A(t)\) is a symmetric matrix. The potential \(F\) satisfies the Carathéodory conditions and has superquadratic growth near infinity and near zero. It has to be remarked that the set of assumptions in this paper enables to treat also problems where the Ambrosetti-Rabinowitz condition may not be satisfied. The main result guarantees the existence of at least one nontrivial solution satisfying \(x(0)=x(b), \, x'(0)=x'(b)\). For the proof, a slight variant of a linking theorem due to \textit{S. Li} and \textit{M. Willem} [J. Math. Anal. Appl. 189, No.~1, 6--32 (1995; Zbl 0820.58012)] is applied.
scientific article; zbMATH DE number 5734958