On the solution structure of nonlinear Hill's equation II, local results

The behavior of solution components in W-2,W-p(IR) x IR of nonlinear Hill's equation near boundary points of the continuous spectrum is discussed in full detail. The starting point of several articles concerned with the study of this solution structure is the existence result of unbounded solution components in a recent paper. The main assumption there is the concentration of the nonlinearity to a compact interval which allows the reduction to an equivalent nonlinear Sturm-Liouville problem with parameter dependent boundary conditions. In particular, this approach makes it possible to analyse the solution structure of the Sturm-Liouville and the original problem. It turns out in the present article that each solution component consists of a branch near the bifurcation point. For the behavior of the solution branches of the original problem, the bifurcation direction and order of bifurcation play an important role. Using the corresponding local results, a class of nonlinearities can be constructed such that at each boundary point of the spectrum infinitely many branches bifurcate asymptotically from infinity.