Bifurcation points of a singular boundary-value problem on (0,1)

Following earlier work on some special cases [11] and [17] and on the analogous problem in higher dimensions [10] and [20], we make a more thorough investigation of the bifurcation points for a nonlinear boundary value problem of the form−{A(x)u′(x)}′=f(λ,x,u(x),u′(x)) for 00 for x∈(0,1] and limx→0⁡A(x)x2=a>0. It was observed in [11] and [17] that if the exponent 2 is replaced by a value less than 2 then classical bifurcation theory can be used to treat the problem. The paper [17] deals with the case f(λ,x,s,t)=λsin⁡sf(λ,x,s,t)=λsin⁡s whereas [11] covers the more general form f(λ,x,s,t)=λF(s)f(λ,x,s,t)=λF(s). Here we admit a much broader class of nonlinearities and some new phenomena appear. In particular, we encounter situations where bifurcation does not occur at a simple eigenvalue of the linearization lying below the essential spectrum.