Bifurcation from the essential spectrum for almost-periodic perturbations of Hill's equation (Q1589874)

Almost-periodic perturbations of Hill's equation \(-\ddot u +Vu-r|u|^{p-1}u=\lambda u\) are considered. Here, \(V\) is periodic and \(r\) almost-periodic. The author proves that (under certain hypotheses) the infimum of the spectrum of the linearisation at \(u=0\) is a bifurcation point of the nonlinear equation. The approach is a variational one. For that reason the problem has been formulated in \(H^1(\mathbb R)\). The main theorem extends a bifurcation result obtained by \textit{C. A. Stuart} [Bull. Belg. Math. Soc. -- Simon Stevin Suppl. (1995; Zbl 0864.47037)].