## Info

In this result, the uniqueness of L{x) is a direct consequence of the fact that any solution of (2) must satisfy (4). All these results where substantially generalized by F. C. Cirstea and V. Radulescu4'5, who proved the following general result.

Theorem 1.2. Suppose f £ C1^, 00) satisfies /(0) = 0, lim 4^2=0 and Lx := lim lik^L £ [0,1]. (5)

Then, (2) has a unique positive solution, L, and

and h(t) is the unique solution to the integral equation

[ (2 f rpdr) 2 ds= f y/f for each t> 0. Jh(t) \ JO J JO

By differentiating (6) with respect to t and integrating reveals that, for each t > 0, but yet no general uniqueness result within the spirit of Theorem 1.1 seems to be available, even in the presence of radial symmetry. Our proof of Theorem 1.1 relies upon (1) and the strong maximum principle, through a re-scaling argument showing that the minimal and the maximal large solutions are equal. Therefore, Theorem 1.1 substantially differs from Theorem 1.2 because, instead of being based upon the boundary blow-up rates of the large solutions, relies upon the monotonicity of /.

Nevertheless, in a further stage, the knowledge of the boundary blowup rates of the large solutions is imperative to use the localization method of J. Lopez-Gomez16 in obtaining general uniqueness results for general problems, either in the absence of radial symmetry, or in the presence of radial symmetry without imposing (1) (cf. J. Lopez-Gomez19 for further details). Consequently, this paper also analyzes the boundary blow-up of the solutions of (2). Our main result concerning this issue can be stated as follows.

Theorem 1.3. Suppose \>0,p>l, and f € C[ 0, oo) satisfies /(0) = 0, (1), and

where

where L is the unique solution of (2).

This result provides us with the boundary blow-up rate of the large solution of (2), in terms of /, in a rather explicit way. A sufficient condition for (8) is given through the following lemma of technical nature.

Lemma 1.1. Suppose p > 1, f £ C[0, oo) satisfies f(0) = 0 and (1), and there exists e > 0 such that:

CI. f S C2(0,e] and, for each t e (0,e], f'{t) > 0 and (Log f)"(t) < 0. C2. The function f'it) f* f f(t)

is non-oscillating in (0,e], in the sense that either B^ (t) > 0 for each t G (0, e], or B'w(t) < 0 for each t e (0,e], or B^ = B0 in (0,e] for some constant Bo-

Then, the limit (8) exists and, actually, Iq € [l,oo).

The local logarithmic concavity of / in condition CI is a rather natural and very weak non-oscillation condition, because, as a consequence of /(0) =0, we have that lim Log/(i) = -oo.

Subsequently, we will consider the function

For each t € (0, e], a satisfies a'(t) = -a2(t) (Log/)"(*)

and, hence, a'(t) > 0 if and only if (Log /)" (i) < 0. Thus, in such case, an := lima(i) > 0

no is well defined. Moreover, by L'Hôpital's rule, lim4^ = 21im^- = 2a0.

Consequently, the local logarithmic concavity of / entails the existence of the first limit of (5). Necessarily, <zo = 0 if /(0) = 0. The mathematical analysis carried out in this work goes beyond revealing that such kind of restrictions on / are only necessary to characterize the boundary blow-up rates of the solutions of (2) through the solution of the associated one-dimensional problem ru" = fvP, t > o,

By Proposition 3.3 (see Section 3), if / € C3(0, e) and a and a' axe non-oscillating in (0, e], then the functions B^ defined in (11) are non-oscillating for every p> 1. Moreover, for any p > 1, lim Bm (t) = . —- if oi := lim a'(t) € [0, oo). UO ai(p+l) + l no w L '

Consequently, as for any t € (0, e] the following is satisfied