Abstract
In this paper, we give the exact asymptotic behavior of the unique positive solution to the following singular boundary value problem
{-1/A(Au')' = P(x)g(u), x is an element of(0,1)
u > 0, in (0,1),
lim(x > 0+)(Au')(x) = 0, u(1) = 0,
where A is a continuous function on [0,1), positive and differentiable on (0,1) such that 1/A is integrable in a neighborhood of 1, g is an element of C-1 ((0,infinity), (0,infinity)) is nonincreasing on (0,infinity) with limt(t > 0)g'(t) integral(t)(0)1/g(s) ds = -C-g <= 0 and p is a nonnegative continuous function in (0,1) satisfying
0 < p(1) = lim inf(x -> 1)p(x)/h(1-x) <= lim sup(x -> 1)p(x)/h(1-x) = p(2) < infinity
where h(t) = ct-lambda exp(integral(eta)(t) z(s)/s ds), lambda <= 2, c > 0 is continuous on [0,eta] for some eta > 1 such that z(0) = 0.