Abstract
In this article, we give an exact behavior at infinity of the unique solution to the following singular boundary value problem
- 1/A(Au')' = q(t)g(u), t is an element of(0, infinity),
u > 0, lim(t -> 0) A(t)u'/(t) = 0, lim(t ->infinity) u(t) = 0.
Here A is a nonnegative continuous function on [0, infinity), positive and differentiable on (0, infinity) such that
lim(t ->infinity) tA'(t)/A(t) = alpha > 1, g is an element of C-1 ((0, infinity), (0, infinity))
is non-increasing on (0, infinity) with lim(t -> 0)g'(t) integral(t)(o) ds/g(s) = -C-g <= 0 and the function q is a nonnegative continuous, satisfying
0 < a(1) = lim inf(t ->infinity) (q(t))(h(t)) <= lim sup (t ->infinity) (q(t))(h(t)) = a(2) < infinity,
where h(t) = ct(-lambda) exp(integral(t)(1) y(s)/sds), lambda >= 2, c > 0 and y is continuous on [1, infinity) such that lim(t ->infinity)y(t) = 0.