Following earlier work on some special cases [17,11] and on the analogous problem in higher dimensions [10,20], we make a more thorough investigation of the bifurcation points for a nonlinear boundary value problem of the form -{A(x)u' (x)}'.= f (lambda, x, u(x), u' (x)) for 0 < x < 1, integral(1)(0) A(x)u' (x)(2)dx < infinity and u(1) = 0, where, for all lambda is an element of R and x is an element of (0, 1), f (lambda, x, 0, 0) = 0 and partial derivative(3)f(lambda, x, 0, 0) = lambda - V (x) and partial derivative(4) f (lambda, x, 0, 0) = 0, so that the formal linearization about a trivial solution u equivalent to 0 is -{A(x)v'(x)}' + V(x)v(x) = lambda v(x) for 0 < x < 1. Even when f is a smooth function of all its variables, standard bifurcation theory does not apply to the problem and the results differ from the usual conclusions. This is because we deal with the case where the coefficient A has a critical degeneracy as x -> 0 in the sense that [GRAPHICS] It was observed in [17,11] 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 (lambda, x, s, t) = lambda sin s whereas [11] covers the more general form f (lambda, x, s, t) = lambda F(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, (C) 2015 Elsevier Inc. All rights reserved.