Homoclinic solutions to periodic motions in a class of reversible equations
The authors deal with the equation \[ \ddot u(t) + W_u(t, u(t)) = h(t), \] where \(W \in C^2 (\mathbb{R}\times \mathbb{R}; \mathbb{R})\) is 1-periodic in \(t\) and \(S\)-periodic in \(u\), \(h \in C (\mathbb{R}; \mathbb{R})\) is 1-periodic and has zero mean, and \(V(t,x) = W(t,x) - h(t)x\) is even in \(t\). They study the existence of homoclinic and heteroclinic solutions to consecutive global minimizers \(u_1\), \(u_2\) of the action functional associated to the 1-periodic solutions. The authors prove: 1. The existence of heteroclinics connecting \(u_1\) and \(u_2\). 2. The existence of infinitely many homoclinic solutions to \(u_1\) and \(u_2\) provided an additional assumption is met. This additional assumption constrains the abundance of heteroclinic solutions, excluding in particular the case of autonomous systems. 3. The nonexistence of homoclinic solutions if there are no consecutive minimizers. 4. Under nondegeneracy of the minimizers, nonexistence of homoclinic solutions when the additional condition in 2 does not hold. As mentioned by the authors, this paper is strongly influenced by the work of P. Rabinowitz, in particular by investigations in [Ergodic Theory Dyn. Syst. 14, No. 4, 817-829 (1994; Zbl 0818.34025) and Differ. Integral Equ. 7, No. 5-6, 1557-1572 (1994; Zbl 0835.34050)].
scientific article; zbMATH DE number 1398356