Exact Boundary Behavior of the Unique Positive Solution for Singular Second-order Differential Equations

In this paper, we give the exact asymptotic behavior of the unique positive solution to the following singular boundary value problem      − 1 A (Au) = p(x)g(u), x ∈ (0, 1), u > 0, in (0, 1), lim x→0 + (Au)(x) = 0, u(1) = 0, where A is a continuous function on [0, 1), positive and differentiable on (0, 1) such that 1 A is integrable in a neighborhood of 1, g ∈ C 1 ((0, ∞), (0, ∞)) is nonincreasing on (0, ∞) with lim t→0 g (t) t 0 1 g(s) ds = −C g ≤ 0 and p is a nonnegative continuous function in (0, 1) satisfying 0 < p 1 = lim inf x→1 p(x) h(1 − x) ≤ lim sup x→1 p(x) h(1 − x) = p 2 < ∞, where h(t) = ct −λ exp(η t z(s) s ds), λ ≤ 2, c > 0 and z is continuous on [0, η] for some η > 1 such that z(0) = 0.


Introduction
In this paper, we give the exact asymptotic behavior near the boundary of the unique positive solution to the following singular problem − 1 A (Au ) = p(x)g(u), x ∈ (0, 1), u > 0, in (0, 1), (1.1) subject to the boundary conditions 2) The functions A, p and g satisfy the following assumptions.
This implies that lim t→0 g(t) satisfy the assumption (H 3 ), as well as the function When A ≡ 1, problems of type (1.1) with various boundary conditions arise in the study of boundary layer equations for the class of pseudoplastic fluids and have been studied for both bounded and unbounded intervals of R (see [4,5,23,27] and the references therein).
When A(t) = t n−1 (n ≥ 2), the operator u → 1 A (Au ) appears as the radial part of the Laplace operator ∆ (see [24]).Our setting includes the scalar curvature equation and the relativistic pendulum equation, which correspond to For various existence, uniqueness and asymptotic behavior results of such problem, we refer the reader to [8-11, 14, 21, 25, 26] and the references therein.However, we emphasize that in problem (1.1) the function A could be singular at t = 1.
On the other hand, the singular nonlinear problem subject to different boundary conditions has been considered by many authors, where A is a continuous function on [0, 1), positive and differentiable on (0, 1) satisfying some appropriate conditions (see for example [1,2,13,16,17,19]).In [15, Theorem 5], Mâagli and Masmoudi investigated equation (1.3) with boundary value conditions u (0) = u(1) = 0.They supposed that f is a nonnegative continuous function on (0, 1) × (0, ∞) and nonincreasing with respect to the second variable.Under some appropriate conditions on the function A, they proved the existence of a unique positive solution u in C([0, 1]) ∩ C 2 ((0, 1)) to (1.3) and gave estimates on such a solution.In particular they extended some results of [1,2] and [19].Our aim in this paper is to establish the exact boundary behavior of the unique solution to problem (1.1)-(1.2).
Note that functions belonging to the class K are in particular slowly varying functions.The theory of such functions was initiated by Karamata in a fundamental paper [12].
We also point out that the first use of the Karamata theory in the study of the growth rate of solutions near the boundary is done in the paper of Cîrstea and Rȃdulescu [7].
Typical examples of functions belonging to the class K (see [3,18,22]) are: and ω is a sufficiently large positive real number such that L is defined and positive on (0, η].
Throughout this paper, we denote by ψ g the unique solution determined by and we mention that lim Our first result is the following.
An immediate consequence of Theorem 1.3 is the following.(a) When C g = 1, then we have Example 1.5.Let g be the function defined by and p be a nonnegative continuous function in (0, 1) satisfying where ).Let u be the unique solution of (1.1)-(1.2).Then, we have the following exact behavior.
(H 5 ) A is a continuous function on [0, 1), positive and differentiable on (0, 1) such that is bounded in a neighborhood of 1 for some where γ ≥ 0 and L ∈ K with η 0 s ds = ∞.Our second result is the following.Theorem 1.6.Assume that hypotheses (H 5 ) and (H 6 ) are fulfilled.Then the problem . The content of this paper is organized as follows.In Section 2, we present some fundamental properties of Karamata regular variation theory.In Section 3, exploiting the results of the previous section, we prove Theorems 1.3 and 1.6 by constructing a convenient pair of subsolution and supersolution.

On the Karamata class K
In this section, we collect some properties of Karamata functions.Proposition 2.1 ([18, 22]).
(i) Let L 1 , L 2 ∈ K and q ∈ R. Then the functions belong to the class K.
(ii) Let L be a function in K and ε > 0.
Then we have lim 18,22]).Let µ ∈ R and L be a function in K defined on (0, η].Then the following hold. (i (ii µ+1 .The proof of the next lemma can be found in [6].
Using the definition of Karamata class and the previous lemmas, we obtain the following.

Lemma 2.6 ([25]
). (i (ii) A positive measurable function f belongs to NRVZ ρ if and only if lim t→0 t f (t) (vi) Let f ∈ NRVZ ρ and m 1 , m 2 be two positive functions on (0, ∞) such that 3 Proofs of Theorems 1.3 and 1.6 In the sequel, we denote by dt, for x ∈ (0, 1), and we let L A u := 1 A (Au ) = u + A A u .Note that since the function A satisfies (H 1 ), the function v 0 (x) is well defined and we have L A v 0 = − 1 A .The following result will play a crucial role in the proof of our main result.Lemma 3.1.Assume (H 1 ), then there exists L 0 ∈ K such that Proof.It is clear that On the other hand, by (H 1 ), we have lim x→1 So by Lemma 2.6, we deduce that the function f (t) := A(1 − t) belongs to NRVZ α .Therefore, there exists L 0 ∈ K such that Hence by using this fact, Proposition 2.1 (i) and Lemma 2.2 (ii), we deduce that Combining (3.2) and (3.3), we obtain the required result.This completes the proof.
Put u = v ds = −C g with C g ≥ 0.

Definition 1 . 1 .
The class K is the set of all Karamata functions L defined on (0, η] by L(t) := c exp

Corollary 1 . 4 .
Let u be the unique solution of problem (1.1)-(1.2).Then, we have the following exact boundary behavior.