| Bifurcation from the essential spectrum for almost-periodic perturbations of Hill's equation (2007) | |||||||||||||
Abstract | |||||||||||||
| We show that the inmum of the essential spectrum of the linearisation of the equation u(x) + V (x)u(x) r(x)ju(x)j p 1 u(x) = u(x) (1) is a bifurcation point for (1). Here the potential V is periodic and the function r almost-periodic. Thus (1) is an almost-periodic perturbation of Hill's equation. Bifurcation results of Kupper and Stuart are combined with existence theory for almost-periodic problems developed by Serra, Tarallo and Terracini. 1 Introduction Many results have been obtained on bifurcation from points in the essential spectra of linearisations of nonlinear equations - for a survey, see [11]. In particular, a series of works initiated by T. Kupper and C.A. Stuart has explored such phenomena for equations such as u(x) + V (x)u(x) r(x)ju(x)j p 1 u(x) = u(x) u 2 H 2 (R) n f0g; 2 R; (2) under the assumptions that 2 < p < 6; V; r 2 L 1 (R); r 0 and V is periodic (see [10] and the references therein). Their approach is variational. Non-zero solutions f... | |||||||||||||
Publication details | |||||||||||||
| |||||||||||||