Some results about positive solutions of a nonlinear equation with a weighted Laplacian

2∗ = 2n n− 2 appears, and it is known that if 1 < q < 2∗, all bounded solutions have a first positive zero, and if q ≥ 2∗, then the solutions are positive in (0,∞). More recently, in 1993, the case of (E) with a weight in the right hand side, B(r) = 1 1+rγ , γ > 0, that is the Matukuma equation, was studied by Ni-Yotsutani [10], Li-Ni [7], [8], [9], and Kawano-Yanagida-Yotsutani [5], where the problem ∗ This research was supported by FONDECYT-1030593 for the first author, Fondap Matemáticas Aplicadas for the second author and FONDECYT-1030666 for the third author.


Introduction
We consider the problem of classification of bounded positive solutions to Here q > 2, and A, B are weight functions, i.e., a.e.positive measurable functions.Many authors have dealt with the non weighted case, i.e., with positive solutions to the equation where q > 2, see for instance [4].
In this case, when n > 2, the critical number 2 * = 2n n − 2 appears, and it is known that if 1 < q < 2 * , all bounded solutions have a first positive zero, and if q ≥ 2 * , then the solutions are positive in (0, ∞).
then there exists a unique α * > 0 such that the solution u(•, α) of (1.1) satisfies -u(r, α) > 0 for all r > 0 with lim Later, in 1995, Yanagida and Yotsutani [11] considered the case of a more general weight in the right hand side, and they studied the problem for K satisfying decreasing and nonconstant in (0, ∞).
They defined the critical numbers From (K 1 ) σ > −2, and then they set and proved the following: Theorem B. Let n > 2 and assume that the weight K satisfies (K 1 ) and (K 2 ).Then (i) If 2 < q ≤ q , then for any α > 0, the solution u(•, α) of (1.2) has a first positive zero in (0, ∞).
(iii) If q < q < q σ , then there exists a unique α * > 0 such that the solution u(•, α) of (1.2) satisfies -u(r, α) > 0 for all r > 0 with lim -u(r, α) > 0 for all r > 0 with lim Clearly, the result in Theorem A is a particular case of that of Theorem B, since 1+r γ satisfies all the assumptions with σ = 0 and = −γ.We will deal here with the case A = B in (P ) when the solutions are radially symmetric: where |x| = r and now the function b(r) := r N −1 B(r) is a positive function satisfying some regularity and growth conditions.We will see in section 3 that under some extra assumption on the weight K in (1.2), the problem considered in [11] is a particular case of ours.
Since we are interested only in positive solutions, we will study the initial value problem Our note is organized as follows: in section 2 we will introduce some necessary conditions to deal with with our problem and we will state our main results which are a particular case of the work in [2].Finally, in section 3 we compare our result with the one given in Theorem B.

Main results
We introduce next some necessary assumptions to deal with (IV P ).We note that if u is a solution to our problem, then for all r > 0, and thus u (r) < 0 for all r > 0. If for some positive R it happens that u(R) = 0, u(r) > 0 for r ∈ (0, R), then for all r ≥ R and such that u(r) ≤ 0, we have that implying that u remains negative for all r ≥ R. If on the contrary it holds that u(r) > 0 for all r > 0, then and thus, for r ≥ s we have and we conclude that 1/b ∈ L 1 (s, ∞) for all s > 0. Putting it in another way, if 1/b ∈ L 1 (1, ∞), then u must have a first positive zero.Therefore, keeping in mind that we are interested in the positive solutions to (P r ), there is no loss of generality in assuming that 1/b ∈ L 1 (s, ∞) for all s > 0.
Moreover, if u is any solution to our problem, then for r ≥ s small enough it holds that b|u and thus b ∈ L 1 (0, 1) is a necessary condition for the existence of solutions to (IV P ).Finally, it can be shown that is necessary and sufficient for the existence and uniqueness of solutions to (IV P ).Hence, our basic assumptions on the weight b will be: By a solution to (IV P ) we understand an absolutely continuous function u defined in the interval [0, ∞) such that b(r)u is also absolutely continuous in the open interval (0, ∞) and satisfies the equation in (IV P ).
We will show that the behavior of function is crucial in the study of solutions to (IV P ).This function played a key role when studying the problem of existence of positive solutions to the corresponding Dirichlet problem associated to our equation, see [1].The behavior at 0 of this function is closely related to the inclusion of weighted Sobolev spaces.For a proper definition of these spaces we refer to Kufner-Opic [6].Now also the behavior at ∞ of this function will be crucial for our classification results.Let us define and put where we set ρ ∞ = ∞ if W = ∅.It can be proved that condition (H 3 ) implies that 2 ∈ U and thus We will prove in section 2 that these critical numbers can be computed as We will denote the unique solution to (IV P ) by u(r, α).As it is standard in the literature, we will say that -u(r, α) is a crossing solution if it has a zero in (0, ∞).
In the case that u is a crossing solution, we will denote its (unique) zero by z(α).
Our main results consist of a classification of the solutions according to the relative position of q with respect to the critical values ρ 0 and ρ ∞ .In these results, the function plays a fundamental role, the connection of this function with the critical values follows since and Also, we note that in the non weighted case, that is, b(r) = r n−1 , n > 2, we have and thus c(r) ≡ 2n n − 2 .
Our first classification result generalizes the non weighted case: Theorem 2.1 Let the weight b satisfy assumptions (H 1 ), (H 2 ) and (H 3 ).Let q > 2 be fixed and assume that c(r (i) If q < ρ * , then u(r, α) a crossing solution for any α > 0.
(ii) If q = ρ * , then u is the rapidly decaying solution given by where C is a positive constant.
Theorem 2.2 Let the weight b satisfy assumptions (H 1 ), (H 2 ) and (H 3 ), and assume that they also satisfy the function r → c(r) is decreasing on (0, ∞).
This result, as well as some very strong generalizations will appear in [2].

Final remarks
In this section, we will compare our result in Theorem 2.2 with Theorem B stated in the introduction.To this end, we will show that if in addition to (K 1 ) and (K 2 ), we assume that then the assumptions in Theorem 2.2 are satisfied.Indeed, as in [3], we make the change of variable and the problem By (3.1), r(0) = 0 and r(∞) = ∞.Next, we will see that assumptions (H 1 ), (H 2 ), and (H 3 ) are satisfied for this b.Clearly, we only need to check that the first in (H 2 ) and (H 3 ) are satisfied.We begin by showing that b ∈ L 1 (0, 1).Indeed, by making the change of variable r = t 0 K 1/2 (τ )dτ , we find that where here and in the rest of this note t 1 is defined by 1 = t1 0 K 1/2 (τ )dτ , and thus b ∈ L 1 (0, 1).Also,   Finally, we will see that under (K 2 ), c is decreasing, and thus our theorem applies: Indeed, it can be seen that in the variable t, Hence, if c (t) > 0 for t ∈ (0, t 0 ), then tc (t) c(t) must decrease in (0, t 0 ).This, together with the fact that lim t→0 tc (t) c(t) = 0, implies that c (t) < 0 in (0, t 0 ), a contradiction.Hence, there are points t > 0 in every interval (0, t 0 ) where c (t) < 0, implying that if c is not always decreasing, it must have a minimum, which is not possible.