Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight

In this article, we prove the existence of eigenvalues for the problem {(phi(p)(u'(t)))' + lambda h(t)phi(p)(u(t)) = 0, t is an element of (0,1), Au(0) - A'u'(0) = 0 Bu(1) + B'u'(1) = 0 under hypotheses that phi(p)(s) = vertical bar S vertical bar Sp-2, p > 1, and h is a nonnegative measurable function on (0, 1), which may be singular at 0 and/or 1. For the result, we establish the existence of connected components of positive solutions for the following problem: {(phi(p)(u'(t)))' + lambda h(t)f(u(t)) = 0, t is an element of (0,1), u(0) = 0 au'(1) +c(lambda, u(1)) = 0, where lambda is a real parameter, a >= 0, f is an element of C((0, infinity), (0, infinity) satisfies inf(s is an element of(0,infinity))f (s) > 0 and limsup(s -> 0)s(alpha)f (s) < infinity for some alpha > 0.