From Section 3 in [4], (IVP) has a unique solution u(r) ~ cZ([0, oo)) with ur(0) = 0, which is denoted by u(r; ~). Moreover, it is easy to see that this solution decreases as long as it is positive. If it stays positive for all r > 0, then we want to determine the precise asymptotic behaviour. In this direction, Peletier et al. [5] have shown a more general result, also valid for solutions with sign changes. THEOREM 1.1 (see [3] and[5]). Let u(r; ~) be a solution of (IVP). Then S := lim r2Xu(r; or) T--~ oo exists and is finite for every c~ > 0. Moreover, if S --0, then there exists a constant R ~ 0 such that (r) u(r; ~) = Rr 2x-" exp -~-I1 + O(r-e)l as r ~ ~.
So for our purpose we classify solutions of (IVP) as follows:
(i) u(r; ~) is a crossing solution if u(r; c~) has a zero in (0, co), i.e. there exists some z e (0, ~) such that u(z; cO = O.
(ii) u(r; a) is a rapidly decaying solution if u(r; oO > 0 in [0, ~) and u(r; cO satisfies S=0.
(iii) u(r; ~) is a slowly decaying solution if u(r; a) > 0 in [0, ~) and u(r; ~) satisfies S>O.
In view of the above classification, the purpose of this paper is to decide whether u(r; c~) is a crossing solution, a rapidly decaying solution or a slowly decaying solution for any initial value c~ e (0, ~). Partial answers are already given by several authors. Summarizing their results, the following can be said: (I) If n >\_ 1, p > 1 and 2 \> n/2, then u(r; a) is a crossing solution for every c~ > 0 (see Weissler [61). (II) Suppose that p is supercritical, i.e. n > 2 and p >\\_ (n + 2)/(n -2). If 0 < 3. < max[l, n/4], then u(r; a) is a slowly decaying solution for every a > 0 (see Atkinson and Peletier [1]). (III) If n \_> 3(n ~ N), p = (n + 2)/(n -2) and max 11, n/4l < ~ < n/2, then there exists a rapidly decaying solution (see Escobedo and Kavian [2]).
(IV) Suppose thatp is subcritical, i.e. 1
2 andp > 1 in case n \_ 2. If 0 < 2 < n/2, then 0 < a. := inf[o~ > 0; u(r; ~) is a crossing solutionJ < oo and u(r; et.) is a rapidly decaying solution. Moreover, u(r; a) is a crossing solution for every sufficiently large a and a slowly decaying solution for every sufficiently small (see [3] and [61). (V) If n = 1, p > 1 and 0 < 2 < 1/2, there exists a unique number a\* e (0, oo) such that u(r; a\) is a rapidly decaying solution, u(r; a) is a crossing solution for every e (or\, oo) and a slowly decaying solution for every o~ ~ (0, a\*) (see [6]). So especially existence and nonexistence of rapidly decaying solutions for subcritical p is well understood. However, the uniqueness remained an open problem except for n = 1. Recently in [4], the second author has proved that a similar result as (V) holds in the subcritical case, if n >\_ 3 and 2 = 0 or 2 = 1. He has shown that there exists a unique positive number o~Γ such that u(.; o~) > 0 in [0, oo) for every o~ e (0, aΓ] and a crossing solution for every et ~ (ax, co). Moreover, u(r; o~Γ) is the most rapidly decaying solution among decaying solutions satisfying u(r; c~Γ) = O(r 2Γ-" exp(-rZ/4)) as r --, oo.