chapter  6
32 Pages

A canonical one-dimensional problem

The main goal of the second part of this book is to obtain general uniqueness results for the large solutions constructed in the first part. The whole program adopted to accomplish this task will be divided into three steps delimited by each of the chapters included in this part. This chapter establishes the existence and the uniqueness of the positive solution, `(x), of the singular boundary value problem{

u′′(x) = a(x)h(u(x)), x > 0, u(0) =∞, u(∞) = 0, (6.1)

under the following conditions on a(x) and h(u):

(A1) a ∈ C[0,∞) satisfies a(x) ≥ a(y) > 0 whenever x ≥ y > 0. (A2) h ∈ C1[0,∞) satisfies h(0) = h′(0) = 0, h′(u) > 0 if u > 0, as well as the

Keller-Osserman condition

I(u) :=

dx√∫ x u h <∞, u > 0, lim

u→∞ I(u) = 0. (6.2)

Moreover, it will ascertain the blow-up rate of `(x) at x = 0 when, besides (A1) and (A2), the next conditions holds.