Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 49, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXACT ASYMPTOTIC BEHAVIOR OF THE POSITIVE
SOLUTIONS FOR SOME SINGULAR DIRICHLET PROBLEMS
ON THE HALF LINE
HABIB MÂAGLI, RAMZI ALSAEDI, NOUREDDINE ZEDDINI
Abstract. In this article, we give an exact behavior at infinity of the unique
solution to the following singular boundary value problem
1
− (Au0 )0 = q(t)g(u), t ∈ (0, ∞),
A
u > 0, lim A(t)u0 (t) = 0,
lim u(t) = 0.
t→∞
t→0
Here A is a nonnegative continuous function on [0, ∞), positive and differentiable on (0, ∞) such that
lim
t→∞
tA0 (t)
= α > 1,
A(t)
g ∈ C 1 ((0, ∞), (0, ∞))
R
is non-increasing on (0, ∞) with limt→0 g 0 (t) 0t
function q is a nonnegative continuous, satisfying
0 < a1 = lim inf
t→∞
ds
g(s)
= −Cg ≤ 0 and the
q(t)
q(t)
≤ lim sup
= a2 < ∞,
h(t)
t→∞ h(t)
R y(s)
where h(t) = ct−λ exp( 1t s ds), λ ≥ 2, c > 0 and y is continuous on [1, ∞)
such that limt→∞ y(t) = 0.
1. Introduction
In this article, we give the exact asymptotic behavior at infinity of the unique
positive solution to the singular problem
1
(Au0 )0 = −q(t)g(u) , t ∈ (0, ∞),
A
(1.1)
u > 0 , in (0, ∞)
lim A(t)u0 (t) = 0,
t→0+
lim u(t) = 0,
t→∞
where the functions A, q and g satisfy the following assumptions.
(H1) A is a continuous function on [0, ∞), positive and differentiable on (0, ∞)
such that
t A0 (t)
lim
= α > 1.
t→∞ A(t)
2010 Mathematics Subject Classification. 34B16, 34B18, 34D05.
Key words and phrases. Singular nonlinear boundary value problems; positive solution;
exact asymptotic behavior; Karamata regular variation theory.
c
2016
Texas State University.
Submitted December 8, 2015. Published February 17, 2016.
1
2
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
EJDE-2016/49
(H2) q is a nonnegative continuous function on (0, ∞) satisfying
tλ q(t)
tλ q(t)
≤ lim sup
= a2 < ∞,
t→∞
L(t)
L(t)
t→∞
R∞
where λ ≥ 2 and L ∈ K (see (1.3) below), such that 1 s1−λ L(s) ds < ∞.
(H3) The function g : (0, ∞) → (0, ∞) is nonincreasing, continuously differentiable such that
Z t
1
0
lim g (t)
ds = −Cg with Cg ≥ 0.
g(s)
t→0+
0
0 < a1 = lim inf
(H4) α + 1 − λ + (λ − 2)Cg > 0.
Using that g is non-increasing, for t > 0, we obtain
Z t
1
0 < g(t)
ds ≤ t.
g(s)
0
Rt 1
This implies limt→0 g(t) 0 g(s)
ds = 0. Now, since for t > 0,
Z t
Z t
Z s
1
1
dr ds = g(t)
ds − t,
g 0 (s)
0 g(s)
0
0 g(r)
we obtain
Z
g(t) t 1
lim
ds = 1 − Cg .
(1.2)
t→0 t
0 g(s)
This implies that 0 ≤ Cg ≤ 1. The functions t−1 log(1 + t), log(log(e + 1t )),
t−ν log(1 + 1t ), exp{(log(1 + 1t ))ν }, ν ∈ (0, 1) satisfy the assumption (H3), as well
as the function
1
t2 e1/t , if 0 < t < ,
2
1
1 2
e , if t ≥ .
4
2
Singular nonlinear boundary value problems appear in a variety of applications
and often only positive solutions are important. When A(t) = 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 [6, 11, 16, 22, 23, 29]) and the references therein.
When A(t) = tn−1 (n ≥ 1), the operator u → A1 (Au0 )0 appears as the radial part of
the laplace operator ∆ (see [10, 30]). Other results of existence and uniqueness of
positive solutions were obtained by Agarwal and O’Regan in [1] on the interval (0, 1)
and in the case where A is continuous on [0, 1], positive and differentiable on (0, 1)
and satisfying an integrability condition. In general the exact asymptotic behavior
of the unique positive solution of (1.1) is extremely complex when the coefficients
are in general continuous functions, even though upper and lower bounds for this
solution are often given (see [1, 4, 10, 15]). Recent research (see [2, 8, 16]) show
that these problems should be studied in the case of Karamata regularly varying
functions. This approach was initiated by Avakumovic [3] and followed by Maric
and Tomic (see [20, 21]). Our aim in this paper is to give a contribution to the
qualitative analysis of problem (1.1) by giving the exact asymptotic behavior at
infinity of the unique positive solution under the previous assumptions on A, q and
g. We note that the existence and uniqueness of such a solution are established by
Mâagli and Masmoudi in [17]. For related results, we refer to Barile and Salvatore
EJDE-2016/49
EXACT ASYMPTOTIC BEHAVIOR
3
[5], Cencelj, Repovš, and Virk [7], Ghergu and Rădulescu [13, 14, 15, 16], Rădulescu
and Repovš [24, 25], Repovš [26, 27].
To state our results, we denote by K the set of Karamata functions L defined on
[1, ∞) by
Z t y(s) ds ,
(1.3)
L(t) := c exp
s
1
where c > 0 and y ∈ C([1, ∞)) such that limt→∞ y(t) = 0.
Remark 1.1. It is clear that a function L is in K if and only if L is a positive
function in C 1 ([1, ∞)) such that
t L0 (t)
= 0.
t→∞ L(t)
lim
(1.4)
Throughout this paper, we denote by ψg the unique solution of the equation
Z ψg (t)
ds
= t , for t ∈ [0, ∞),
(1.5)
g(s)
0
and we mention that
lim tg 0 (ψg (t)) = −Cg .
(1.6)
t→0
Theorem 1.2. Assume (H1)–(H4). Then problem (1.1) has a unique solution
u ∈ C 2 ((0, ∞)) ∩ C([0, ∞)) satisfying
(i) If λ > 2,
ξ 1−Cg
ξ 1−Cg
u(x)
u(x)
1
2
≤ lim inf
≤ lim sup
≤
,
2−λ
2−λ
t→∞ ψg (t
λ−2
L(t))
L(t))
λ−2
t→∞ ψg (t
where ξi =
(ii) If λ = 2,
ai
α+1−λ+(λ−2)Cg
ξ1 1−Cg ≤ lim inf
t→∞
for i ∈ {1 2}.
u(t)
u(t)
≤ lim sup
≤ ξ2 1−Cg
R∞
R ∞ L(s)
t→∞ ψg (
ψg ( t L(s)
ds)
ds)
s
s
t
An immediate consequence of Theorem 1.2 is the following result.
Corollary 1.3. Let u be the unique solution of (1.1). Then, we have the following
exact asymptotic behavior:
(a) When Cg = 1, we have
u(t)
= 1 if λ > 2;
ψg (t2−λ L(t))
u(t)
R
limt→∞
= 1 if λ = 2.
L(s)
ψg ( t∞ s ds)
(i) limt→∞
(ii)
(b) When Cg < 1 and a1 = a2 = a0 , we have
u(t)
a0
= ( (λ−2)(α+1−λ+(λ−2)C
)1−Cg
ψg (t2−λ L(t))
g)
u(t)
a0 1−Cg
R
limt→∞
= ( α−1
)
if λ = 2.
L(s)
ψg ( t∞ s ds)
(i) limt→∞
(ii)
if λ > 2;
Remark 1.4. In the hypothesis (H3), we do not need the monotonicity of the
function g on (0, ∞), but only the fact that g is non-increasing in a neighborhood
of zero.
4
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
EJDE-2016/49
Example 1.5. Let g be the function
(
t2 e1/t , if 0 < t < 12 ,
g(t) = 1 2
if t ≥ 12 .
4e ,
and let q be a nonnegative function in (0, ∞) such that
q(t)
= b0 ∈ (0, ∞),
h(t)
R∞
where h(t) = t−λ L(t), λ ≥ 2 and L ∈ K such that 1 s1−λ L(s) ds < ∞. Then, we
−1
for ξ ∈ (0, e−2 ). Let u be the unique solution of
have Cg = 1 and ψg (ξ) = log(ξ)
(1.1), then we have the following exact behavior:
(i) limt→∞ u(t) log t2−λ1L(t) = 1 if λ > 2;
1
(ii) limt→∞ u(t) log R ∞ L(s)
= 1 if λ = 2.
lim
t→∞
ds
s
t
To establish our second result, we consider the special case where g(t) = t−γ
γ
with γ ≥ 0, and λ = α + 1 + γ(α − 1). Note that in this case Cg = γ+1
and
(α + 1 − λ) + (λ − 2)Cg = 0. We assume that A and q satisfy the following
hypotheses:
(H5) A is a continuous function on (0, ∞) such that A(t) = tα B(t) with α > 1
ν 0
B (t)
and t B(t)
is bounded for t large and ν ∈ (0, 1).
(H6) q is a nonnegative continuous function in (0, ∞) and satisfies
0 < a1 = lim inf
t→∞
q(t)
tγ−1−α(γ+1) L(t)
where L ∈ K with
R∞
1
L(s)
s
≤ lim sup
t→∞
q(t)
tγ−1−α(γ+1) L(t)
= a2 < ∞,
ds = ∞.
Theorem 1.6. Assume (H5), (H6) are satisfied. Then the Dirichlet problem
1
− (A u0 )0 = q(t)u−γ , t ∈ (0, ∞),
A
lim+ A(t)u0 (t) = 0, lim u(t) = 0,
(1.7)
t→∞
t→0
has a unique solution u ∈ C([0, ∞)) ∩ C 2 ((0, ∞)), satisfying
1
(γ + 1)a 1+γ
1
α−1
≤ lim inf
t→∞
t1−α
≤ lim sup
t→∞
≤(
t1−α
u(t)
R t L(s)
1
s
1
ds 1+γ
u(t)
R t L(s)
1
s
1
ds 1+γ
1
(γ + 1)a2 1+γ
)
,
α−1
In particular if a1 = a2 , then
lim
t→∞ 1−α
t
u(t)
Rt
L(s)
1 s
1
ds 1+γ
=
1
(γ + 1)a 1+γ
1
α−1
.
EJDE-2016/49
EXACT ASYMPTOTIC BEHAVIOR
5
2. On the Karamata class
To make the paper self-contained, we begin this section by recapitulating some
properties of Karamata regular variation theory. The following result is due to
[19, 28].
Lemma 2.1. (i) Let L ∈ K and ε > 0, then
lim t−ε L(t) = 0.
t→∞
(ii) Let L1 , L2 ∈ K and p ∈ R. Then L1 + L2 ∈ K, L1 L2 ∈ K and Lp1 ∈ K .
Applying Karamata’s theorem (see [19, 28]), we get the following result.
Lemma 2.2. Let γ ∈ R, L be a function in K defined on [1, ∞). We have
R∞
(i) If γ < −1, then 1 sγ L(s)ds converges. Moreover
Z ∞
t1+γ L(t)
sγ L(s)ds ∼t→∞ −
.
γ+1
t
R∞
(ii) If γ > −1, then 1 sγ L(s)ds diverges. Moreover
Z t
t1+γ L(t)
.
sγ L(s)ds ∼t→∞
γ+1
1
Lemma 2.3 ([8, 18]). Let L ∈ K be defined on [1, ∞). Then
lim R t
t→∞
If further
R∞
1
L(s)
s ds
L(t)
L(s)
ds
1 s
= 0.
(2.1)
= 0.
(2.2)
converges, then
L(t)
lim R ∞ L(s)
t→∞
t
s
ds
Remark 2.4. Let L ∈ K, then using Remark 1.1 and (2.1), we deduce that
Z t
L(s)
t→
ds ∈ K.
s
1
R∞
R ∞ L(s)
If further 1 L(s)
s ds converges, then t → t
s ds ∈ K.
Definition 2.5. A positive measurable function k is called normalized regularly
varying at infinity with index ρ ∈ R and we write k ∈ N RV Iρ if k(s) = sρ L(s) for
s ∈ [1, ∞) with L ∈ K.
Using the definition of the class K and the above Lemmas we obtain the following
lemma.
ρ
Lemma 2.6 ([2]).
(i) If k ∈ N RV Iρ , then limt→∞ k(ξt)
k(t) = ξ , uniformly for
ξ ∈ [c1 , c2 ] ⊂ (0, ∞).
(ii) A positive measurable function k belongs to the class N RV Iρ if and only if
0
(t)
limt→∞ tkk(t)
= ρ.
R∞
(iii) Let L ∈ K and assume that 1 s1−λ L(s) ds < ∞. Then the function
R ∞ 1−λ
θ(t) = t s
L(s) ds belongs to N RV I(2−λ) .
(iv) The function ψg ◦ θ ∈ N RV I(2−λ)(1−Cg ) .
(v) Let m1 , m2 be positive functions on (0, ∞) such that limt→∞ m1 (t) =
ψg (m1 (t))
1 (t)
limt→∞ m2 (t) = 0 and limt→∞ m
m2 (t) = 1. Then limt→∞ ψg (m2 (t)) = 1.
6
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
EJDE-2016/49
3. Proofs of Theorems 1.2 and 1.6
In the sequel, we denote by
Z
v0 (t) =
t
∞
1
ds for t ∈ (0, ∞).
A(s)
Since the function A satisfies (H1), then using Definition 2.5 and assertion (ii) of
Lemma 2.6, we deduce that there exists L0 ∈ K such that A(t) = tα L0 (t), for
t > 1. Hence, using Lemma 2.1, we deduce that 1/A is integrable near infinity. So
the function v0 is well defined, and by Lemma 2.2 we have
Z ∞
1
t1−α
ds ∼
as t → ∞.
(3.1)
v0 (t) =
A(s)
(α − 1) L0 (t)
t
In the sequel, we denote also by LA u :=
LA v0 = 0.
1
0 0
A (Au )
Proof of Theorem 1.2. Let ε ∈ (0, a1 /2). Put
ai
ξi =
(α + 1 − λ) + (λ − 2)Cg
= u00 +
A0 0
Au
and we remark that
for i ∈ {1, 2},
τ1 = ξ1 − ε aξ11 and τ2 = ξ2 + ε aξ22 . Clearly, we have ξ21 < τ1 < τ2 <
R∞
θ(t) = t s1−λ L(s) ds and put
Z ∞
ωi (t) = ψg τi
s1−λ L(s) ds = ψg (τi θ(t)), for t > 0.
3
2 ξ2 .
Let
t
By a simple calculus, for i ∈ {1, 2} we obtain
LA ωi (t) + q(t)g(ωi (t))
h
= g(ωi (t))t−λ L(t)) τi (τi t2−λ L(t)g 0 (ωi (t)) + (λ − 2)Cg )
t A0 (t)
t L0 (t) −α+
− τi ((α + 1 − λ + (λ − 2)Cg ) + ai
− τi
A(t)
L(t)
q(t)
i
+ −λ
− ai .
t L(t)
So, for the fixed ε > 0, there exists Mε > 1 such that for t > Mε and i ∈ {1, 2}, we
have
t A0 (t)
t L0 (t) ε
t L0 (t) 3 t A0 (t)
≤ ,
−α+
− α + τi ≤ ξ2
A(t)
L(t)
2
A(t)
L(t)
4
ε
a(t)
ε
a1 − ≤ −µ
≤ a2 +
2
t L(t)
2
3
ε
|τi (τi t2−λ L(t)g 0 (ωi (t)) + (λ − 2)Cg )| ≤ ξ2 |τi t2−λ L(t)g 0 (ωi (t)) + (λ − 2)Cg | ≤ .
2
4
Indeed, the last inequality follows from (1.6) and the fact that from Lemmas 2.2
and 2.3, we have
t2−λ L(t)
lim R ∞ 1−λ
= 2 − λ,
t→∞
s
L(s) ds
t
for all λ ≥ 2. This implies that for each t > Mε , we have
LA ω1 (t) + q(t)g(ω1 (t)) ≥ g(ω1 (t))t−λ L(t)[−ε + a1 − τ1 ((α + 1 − λ) + (λ − 2)Cg )] = 0
EJDE-2016/49
EXACT ASYMPTOTIC BEHAVIOR
7
and
LA ω2 (t) + q(t)g(ω2 (t)) ≤ g(ω2 (t))t−λ L(t)[ε + a2 − τ2 ((α + 1 − λ) + (λ − 2)Cg )] = 0.
Let u ∈ C 2 ((0, ∞)) ∩ C([0, ∞)) be the unique solution of (1.1) (see [17]). Then,
there exists B > 0 such that
ω1 (Mε ) − B v0 (Mε ) ≤ u(Mε ) ≤ ω2 (Mε ) + B v0 (Mε ) .
(3.2)
We claim that
ω1 (t) − B v0 (t) ≤ u(t) ≤ ω2 (t) + B v0 (t)
for all t > Mε .
(3.3)
Assume for instance that the right inequality of (3.3) is not true. Then the function
h(t) = u(t) − ω2 (t) − B v0 (t) for t > Mε is not negative. Consequently, there exists
t1 > Mε such that h(t1 ) = maxMε ≤t<∞ h(t) > 0. Since h is continuous on [Mε , ∞),
h(Mε ) ≤ 0 and limt→∞ h(t) = 0, then h0 (t1 ) = 0 and h(t) > 0 for t ∈ (t1 − δ, t1 + δ)
for some δ > 0, sufficiently small. Namely u(t) > ω2 (t)+B v0 (t) for t ∈ (t1 −δ, t1 +δ).
Since g is non-increasing on (0, ∞), then
1
1
(A(t)h0 (t))0 = −q(t)g(u(t)) −
(A(t)ω20 (t))0 ≥ q(t)(g(ω2 (t)) − g(u(t))) ≥ 0,
A(t)
A(t)
for t ∈ (t1 − δ, t1 + δ). Which implies h0 (t) ≤ h0 (t1 ) = 0 for t ∈ (t1 − δ, t1 ) and
h0 (t) ≥ h0 (t1 ) = 0 for t ∈ (t1 , t1 + δ). This implies that h has a local minimum at
t1 . Which contradicts the fact that h a global maximum at t1 on [Mε , ∞). This
proves that
u(t) ≤ ω2 (t) + B v0 (t) for all t > Mε .
Similarly, we show that
ω1 (t) − B v0 (t) ≤ u(t)
for all t > Mε .
This proves (3.3).
Now, since ψg ◦ θ ∈ N RV I(2−λ)(1−Cg ) , there exists L̂ ∈ K such that ψg ◦ θ =
(2−λ)(1−Cg )
t
L̂(t) for t ∈ [1, ∞). Moreover since (α − 1) − (λ − 2)(1 − Cg ) > 0, it
follows by Lemma 2.1 that
lim
t1−α
t→∞ t(2−λ)(1−Cg ) L̂(t)
= 0.
This implies that
t1−α
t1−α
ψg (θ(t)) t1−α
R∞
=
lim
=
lim
=0
t→∞ ψg (τi
s1−λ L(s) ds) t→∞ ψg (τi θ(t)) t→∞ ψg (τi θ(t)) ψg (θ(t))
t
lim
uniformly in τi ∈ [ ξ21 , 32 ξ2 ] ⊂ (0, ∞). This together with (3.1) implies
v0 (t)
v0 (t)
= lim
= 0.
t→∞ ψg (τ1 θ(t))
t→∞ ψg (τ2 θ(t))
lim
So, we obtain
u(t)
u(t)
≤ 1 ≤ lim inf
.
t→∞ ω1 (t)
ω2 (t)
Using this fact and assertions (i) and (iv) of Lemma 2.6, we deduce that
lim sup
t→∞
lim inf
t→∞
u(t)
u(t) ω1 (t)
ψg (τ1 θ(t))
= lim inf
≥ lim
= τ1 1−Cg .
t→∞ ω1 (t) ψg (θ(t))
t→∞ ψg (θ(t))
ψg (θ(t))
8
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
EJDE-2016/49
By letting ε approach zero, we obtain
ξ1 1−Cg ≤ lim inf
t→∞
u(t)
.
ψg (θ(t))
Similarly, we obtain
u(t)
≤ ξ2 1−Cg .
ψg (θ(t))
This proves in particular the exact behavior at infinity in the case λ = 2. Now,
2−λ
for λ > 2, we have by Lemma 2.2 that θ(t) ∼t→∞ tλ−2 L(t). Hence it follows by
assertions (i), (iv) and (v) of Lemma 2.6 that for λ > 2, we have
lim sup
t→∞
lim
t→∞
ψg (θ(t))
ψg ((λ − 2)θ(t))
1
ψg (θ(t))
.
= lim
=
ψg ((t)2−λ L(t)) t→∞ ψg ((λ − 2)θ(t)) ψg ((t)2−λ L(t))
(λ − 2)1−Cg
This achieves the proof of the Theorem.
γ
.
Proof of Theorem 1.6. We recall that g(t) = t−γ , λ = α+1+(α−1)γ and Cg = 1+γ
a1
Let ε ∈ (0, 2 ) and put τ1 = (γ + 1)(a1 − ε) and τ2 = (γ + 1)(a2 + ε). Put
Rt
k(t) = 1 L(s)
s ds and
Z ∞
1
1+γ
ωi (t) = (1 + γ)τi
for i ∈ {1, 2},
s1−λ k(s) ds
t
where L is the function given in hypothesis (H6 ). Then, by a simple computation,
we have
LA ωi (t) + q(t)g(ωi (t))
h k(t)
γ = g(ωi (t))t−λ L(t) τi
(τi t2−λ k(t)g 0 (ωi ) + (λ − 1 − α)) −
L(t)
γ+1
q(t)
i
γ
k(t) t A0 (t)
− τi
−α +
τi − τi + ai + −λ
− ai
L(t) A(t)
γ+1
t L(t)
h k(t)
γ = g(ωi (t))t−λ L(t) τi
(τi t2−λ k(t)g 0 (ωi ) + (α − 1)γ) −
L(t)
γ+1
q(t)
i
k(t) t B 0 (t)
τi
− τi
−
+ ai + −λ
− ai .
L(t) B(t)
γ+1
t L(t)
Since g(t) = t−γ and λ = α + 1 + (α − 1)γ, integrating by parts, we obtain
τi t2−λ k(t)g 0 (ωi (t)) + (α − 1)γ
= −γτi t2−λ k(t)(ωi (t))−(1+γ) + (α − 1)γ
t2−α k(t)
R∞
= γ (α − 1) −
(γ + 1) t s1−λ k(s) ds
(α − 1)(1 + γ) R ∞ s1−λ k(s)ds − t2−λ k(t) tR
=γ
∞
(1 + γ) t s1−λ k(s)ds
R ∞ 1−λ
s
L(s) ds
γ
Rt ∞
=
.
1−λ
γ+1 t s
k(s)ds
This gives
k(t)
γ
(τi t2−λ k(t)g 0 (ωi (t)) + (α − 1)γ) −
L(t)
γ+1
EJDE-2016/49
EXACT ASYMPTOTIC BEHAVIOR
γ h
=
γ+1
9
R∞
i
s1−λ L(s) ds
t2−λ k(t)
R∞
−
1
.
2−λ
t
L(t)
s1−λ k(s)ds
t
t
This together with Lemma 2.2 and the fact that k and L are in K, implies
lim
t→∞
tν B 0 (t)
B(t)
t1−ν k(t)
limt→∞ L(t) =
Now since
k(t)
γ
(τi t2−λ k(t)g 0 (ωi (t)) + (α − 1)γ) −
= 0.
L(t)
γ+1
is bounded for t large and by Lemma 2.1, we have
k
L
∈ K and
0, we deduce that
t1−ν k(t) tν B 0 (t) k(t) t B 0 (t) = lim
= 0.
t→∞
t→∞ L(t)
B(t)
L(t)
B(t)
lim
So, for the fixed ε > 0, there exists Mε > 1 such that for t ≥ Mε , we have
hε ε
τ2
εi
LA ω2 (t) + q(t)g(ω2 (t)) ≤ g(ω2 (t))t−λ L(t) + −
+ a2 +
= 0,
3 3 γ+1
3
i
τ1
ε
ε ε
+ a1 −
= 0.
LA ω1 (t) + q(t)g(ω1 (t)) ≥ g(ω1 (t))t−µ L(t)[− − −
3 3 γ+1
3
Let u ∈ C([0, ∞)) ∩ C 2 ((0, ∞)) be the unique solution of (1.7). As in the proof of
Theorem 1.2, we choose C > 0 such that
ω1 (t) − Cv0 (t) ≤ u(t) ≤ ω2 (t) + Cv0 (t)
for t ≥ Mε .
Moreover, thanks to (H6), we have limt→∞ k(t) = ∞. So, using Lemma 2.2, we
obtain
1
1
lim
= lim
1
1
1+γ
R∞
t→∞
t→∞ α−1 (2−λ) τ1 k(t) 1+γ
t
t
tα−1 (1 + γ)τ1 t s1−λ k(s) ds
α−1
1
α − 1 1+γ
= lim
= 0.
t→∞ τ1 k(t)
This and (3.1) gives limt→∞
we have
v0 (t)
ω1 (t)
lim sup
t→∞
= 0. Similarly, we obtain limt→∞
u(t)
u(t)
≤ 1 ≤ lim inf
.
t→∞
ω2 (t)
ω1 (t)
This implies that
lim inf t→∞
u(t)
(1 + γ)
R∞
t
1
≥ τ11+γ .
1
1+γ
s1−λ k(s) ds
Now, as ε tends to zero, we obtain
lim inf t→∞
(1 + γ)
R∞
t
u(t)
1
u(t)
1
≥ ((γ + 1)a1 ) 1+γ .
1
1+γ
1−λ
s
k(s) ds
Similarly, we obtain
lim sup t→∞
(1 + γ)
R∞
t
≤ ((γ + 1)a2 ) 1+γ .
1
1+γ
s1−λ k(s) ds
v0 (t)
ω2 (t)
= 0. So
10
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
Now, since (γ+1)
that
R∞
t
s1−λ k(s) ds ∼t→∞
1
(γ + 1)a 1+γ
1
α−1
t2−λ k(t)
α−1
t→∞
t1−α
u(t)
R t L(s)
≤ lim sup
t→∞
≤(
t(1−α)(γ+1)
α
=
≤ lim inf
EJDE-2016/49
t1−α
s
1
s
1
L(s)
s
ds, we deduce
1
ds 1+γ
u(t)
R t L(s)
1
Rt
1
ds 1+γ
1
(γ + 1)a2 1+γ
)
.
α
In particular, if a1 = a2 , we obtain
lim
t→∞ 1−α
t
u(t)
R t L(s)
s
1
1 =
ds 1+γ
1
(γ + 1)a 1+γ
1
α−1
.
4. Applications
Application 1. We consider the Dirichlet problem
1
β
− (Au0 )0 + (u0 )2 = q(t)g(u) ,
A
u
u > 0 , in (0, ∞) ,
lim A(t)u0 (t) = 0
t→0+
t ∈ (0, ∞),
(4.1)
lim u(t) = 0,
t→∞
where β < 1 and limt→∞ t−λq(t)
= a0 > 0 with λ ≥ 2 and L ∈ K with
L(t)
R ∞ 1−λ
s
L(s) ds < ∞.
1
We assume that A satisfies (H1) and g satisfies the following hypotheses:
(A1) The function t → t−β g(t) is non-increasing from (0, ∞) into (0, ∞).
Rt 1
β
ds = −Cg with max(0, β−1
) ≤ Cg ≤ 1.
(A2) limt→0 g 0 (t) 0 g(s)
(A3) (α − 1) − (λ − 2)(1 − β)(1 − Cg ) > 0
Note that for γ > 0 and −γ < β < 1, the function g(t) = t−γ satisfies (A1 ) and
1
(A2). Put u = v 1−β . Then v satisfies
−β
1
1
− (Av 0 )0 = (1 − β)q(t)g(v 1−β )v 1−β ,
A
v > 0 , in (0, ∞) ,
0
lim A(t)v (t) = 0,
t→0+
1
t ∈ (0, ∞),
(4.2)
lim v(t) = 0,
t→∞
β
The function f (r) = (1 − β)g(r 1−β )r− 1−β is non-increasing on (0, ∞) and a simple
1
computation shows that ψg = (ψf ) 1−β and
Z r
1
0
lim f (r)
ds = (1 − β)(1 − Cg ) − 1 =: −Cf , with 0 ≤ Cf ≤ 1.
r→0
f
(s)
0
Applying Corollary 1.3 to problem (4.2), we deduce that there exists a unique
solution v to (4.2) such that
(a) When Cf = 1, we have
(i) limt→∞
v(t)
ψf (t2−λ L(t))
= 1 if λ > 2;
EJDE-2016/49
EXACT ASYMPTOTIC BEHAVIOR
(ii) limt→∞
ψf
v(t)
R ∞ L(s)
s
t
ds
11
= 1 if λ = 2;
(b) When Cf < 1, we have:
v(t)
a0
]1−Cf
= [ α+1−λ+(λ−2)C
ψf (t2−λ L(t))
f
v(t)
a0 1−Cf
R
limt→∞
= [ α−1
]
if λ = 2.
L(s)
ψf ( t∞ s ds)
(i) limt→∞
(ii)
if 2 < λ < 2 +
α−1
(1−β)(1−cg ) ;
This implies that problem (4.1) has a solution u ∈ C([0, ∞))∩C 2 ((0, ∞)) satisfying
the following exact behavior
(a) When Cg = 1, we have:
(i) if λ > 2, then
lim
t→∞
u(t)
= 1;
ψg (t2−λ L(t))
(ii) if λ = 2, then
lim
t→∞
ψg (
u(t)
R ∞ L(s)
t
s
= 1;
ds)
β
(b) If max(0, β−1
) ≤ Cg < 1, then:
(i) if 2 < λ < 2 +
lim
t→∞
ψg
α−1
(1−β)(1−Cg ) ,
then
u(t)
a0
=[
]1−Cg
L(t))
α − 1 − (λ − 2)(1 − β)(1 − Cg )
(t2−λ
(ii) if λ = 2, then
u(t)
a0 1−Cg
=[
]
.
R ∞ L(s)
α−1
|x|→∞ ψ (
g t
s ds)
lim
Application 2. In this subsection, we assume that the function A satisfy the
following hypothesis
(A4) A is a continuous function on [0, ∞), positive and differentiable on (0, ∞)
A0 (t)
such that A1 is integrable near 0 and limt→∞ t A(t)
= σ ∈ R − {1}.
We are interested in the exact behavior at infinity of the unique positive solution
of the problem
1
(A(t)u0 (t))0 = −p(t)u−γ , t ∈ (0, ∞),
A(t)
u > 0 , in (0, ∞) ,
(4.3)
u(t)
u(0) = 0 , lim
= 0,
t→∞ ρ(t)
R t ds
where γ > 0 and ρ(t) = 0 A(s)
. Let u(t) = ρ(t) v(t) and B(t) = A(t)ρ2 (t) for
t ∈ [0, ∞). Then u is a positive solution of (4.3) if and only if v is a positive
solution of the problem
1
p(t)
(B(t)v 0 (t))0 = −
v −γ ,
B(t)
(ρ(t))γ+1
v > 0 , in (0, ∞) ,
lim B(t)v 0 (t) = 0 ,
t→0+
t ∈ (0, ∞),
lim v(t) = 0 .
t→∞
(4.4)
12
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
EJDE-2016/49
First, we claim that if A satisfies (A4), then
tB 0 (t)
= 1 + |σ − 1| > 1.
t→∞ B(t)
lim
0
(4.5)
0
B (t)
A (t)
Since t B(t)
= t A(t)
+ A(t)2tρ(t) and by Definition 2.5 and assertion (ii) of Lemma
2.6, we have A(t) = tσ L0 (t) for t ≥ a > 1 with L0 ∈ K, then we deduce from
Lemma 2.2 that
• For σ < 1, we have ρ(∞) = ∞ and so
Z t
Z a
1
1 t1−σ
ds
+
ds
∼
as t → ∞.
ρ(t) =
σ
1 − σ L0 (t)
a s L0 (s)
0 A(s)
So
2t
L0 (t) 2t
∼ (1 − σ) 1−σ σ
= 2(1 − σ) as t → ∞.
A(t) ρ(t)
t
t L0 (t)
Consequently in this case we have
t B 0 (t)
= σ + 2(1 − σ) = 2 − σ = 1 + |σ − 1|.
lim
t→∞ B(t)
R ∞ ds
• For σ > 1, we have ρ(∞) = 0 A(s)
ds < ∞. So
2t
2t
∼ σ
→0
A(t) ρ(t)
t L0 (t)ρ(∞)
as t → ∞.
In this case we have
t B 0 (t)
= σ = 1 + |σ − 1|.
t→∞ B(t)
This proves (4.5). Taking into account this fact, we assume that the function p
satisfies the following hypotheses
(A5) p is a nonnegative continuous function (0, ∞) satisfying
lim
tλ p(t)
< ∞,
t→∞ L(t)(ρ(t))γ+1
R∞
where λ ≥ 2 and L ∈ K such that 1 s1−λ L(s) ds < ∞.
γ
(A6) 2 + |σ − 1| − λ + (λ − 2) γ+1 > 0.
0 < a0 = lim
Assume that A and p satisfy (A4)–(A6) and let v be the unique positive solution
of problem (4.4). Then v has the following exact behavior at infinity
(i) if λ > 2, then
1
h
i 1+γ
v(t)
a0
lim
=
.
1
γ
t→∞ [(γ + 1)t2−λ L(t)] 1+γ
(λ − 2)(2 + |σ − 1| − λ + (λ − 2) γ+1 )
(ii) if λ = 2, then
lim
t→∞
[(γ + 1)
v(t)
R ∞ L(s)
t
s
ds]
1
1+γ
=[
1
a0 i 1+γ
|σ − 1|
Consequently, the unique positive solution u of problem (4.3) has the following
exact behavior at infinity
(i) if λ > 2, then
1
h
i 1+γ
u(t)
a0
lim
=
.
1
γ
t→∞ ρ(t)[(γ + 1)t2−λ L(t)] 1+γ
(λ − 2)(2 + |σ − 1| − λ + (λ − 2) γ+1
)
EJDE-2016/49
EXACT ASYMPTOTIC BEHAVIOR
13
ii) if λ = 2, then
u(t)
R
t→∞ ρ(t)[(γ + 1) ∞
t
lim
1
L(s)
1+γ
s ds]
=
h
1
a0 i 1+γ
|σ − 1|
Acknowledgements. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (253-662-1436G). The authors, therefore, acknowledge with thanks DSR technical and financial
support.
The three authors have contributed equally for this article.
References
[1] R. P. Agarwal, D. O’Regan; Singular problems modelling phenomena in the theory of pseudoplastic fluids, ANZIAM J. 45 (2003), 167-179.
[2] R. Alsaedi, H. Mâagli, N. Zeddini; Exact behavior of the unique positive solution to some
singular elliptic problem in exterior domains, Nonlinear Anal. 119 (2015), 186-198.
[3] V. Avakumovic; Sur l’équation différentielle de Thomas-Fermi, Acad. Serbe Sci., Publ. Inst.
Math. 1 (1947), 101-113.
[4] I. Bachar, H. Mâagli, N. Zeddini; Estimates on the Green function and existence of positive
solutions of nonlinear singular elliptic equations, Comm. Contemporary Math. 5 (2003), 401434.
[5] S. Barile, A. Salvatore; Existence and multiplicity results for some Lane-Emden elliptic systems: subquadratic case, Adv. Nonlinear Anal. 4 (2015), 25-35.
[6] A. Callegari, A. Nachman; Some singular, nonlinear differential equations arising in boundary
layer theory, J. Math. Anal. Appl. 64 (1978), 96-105.
[7] M. Cencelj, D. Repovš, Z. Virk; Multiple perturbations of a singular eigenvalue problem,
Nonlinear Anal. 119 (2015), 37-45.
[8] R. Chemmam, A. Dhifli, H. Mâagli; Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems, Electronic Journal Differential Equations,
Vol. 2011 (2011), No. 88, pp. 1-12.
[9] J. S. Fulks, J. S. Maybee; A singular nonlinear elliptic equation, Osaka J. Math. 12 (1960),
1-19.
[10] J. A. Gatica, G. E. Hernandez, P. Waltman; Radially symmetric positive solutions for a class
of singular elliptic equations, Proceedings of the Edinburgh Mathematical Society 33 (1990),
169-180.
[11] J. A. Gatica, V. Oliker, P. Waltman; Singular nonlinear boundary value problems for second
order ordinary differential equations, J. Differential Equations 79 (1989), 62-78.
[12] P. G. de Gennes; Wetting: statics and dynamics, Review of Modern Physics 57 (1985),
827-863.
[13] M. Ghergu, V. D. Rădulescu; Bifurcation and asymptotics for the Lane-Fowler equations, C.
R. Acad. Sci. Paris Ser. I 337 (2003), 259-264.
[14] M. Ghergu, V. D. Rădulescu; Multi-parameter bifurcation and asymptotics for the singular
Lane-Emden-Fowler equation with a convection term, Proc. Roy. Soc. Edinburgh Sect. A 135
(2005), 61-83.
[15] M. Ghergu, V.D. Rădulescu; Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford University Press, 2008.
[16] M. Ghergu, V.D. Rădulescu; Nonlinear PDEs: Mathematical Models in Biology, Chemistry
and Population Genetics, Springer Monographs in Mathematics, Springer-Verlag, Heidelberg,
2012.
[17] H. Mâagli, S. Masmoudi; Sur les solutions d’un operateur différentiel singulier semilinearéaire, Potential Analysis 10 (1999), 289-304.
[18] H. Mâagli, N. Mhadhebi, N. Zeddini; Existence and exact asymptotic behavior of positive
solutions for a fractional boundary value problem, Abstract and Applied Analysis, Volume
2013, Article ID 420514, 6 p.
[19] V. Maric; Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726,
Springer-Verlag, Berlin, 2000.
14
H. MÂAGLI, R. ALSAEDI, N. ZEDDINI
EJDE-2016/49
[20] V. Maric, M. Tomic; Asymptotic properties of solutions of the equation y 00 = f (x)φ(y), Math.
Z. 149 (1976), 261-266.
[21] V. Maric, M. Tomic; Regular variation and asymptotic properties of solutions of nonlinear
differential equations, Publ. Inst. Math. (Belgr.) 21 (1977), 119-129.
[22] A. Nachman, A. Callegari; A nonlinear singular boundary value problem in the theory of
pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275-281.
[23] D. O’Regan; Singular nonlinear differential equations on the half line, Topological Methods
in Nonlinear Analysis 8 (1996), 137-159.
[24] V. Rădulescu, D. Repovš; Perturbation effects in nonlinear eigenvalue problems, Nonlinear
Anal. 70 (2009), 3030-3038.
[25] V. Rădulescu, D. Repovš; Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Chapman and Hall/CRC, Taylor & Francis Group,
Boca Raton, FL, 2015.
[26] D. Repovš; Singular solutions of perturbed logistic-type equations, Appl. Math. Comp. 218
(2011), 4414-4422.
[27] D. Repovš; Asymptotics for singular solutions of quasilinear elliptic equations with an absorption term, J. Math. Anal. Appl. 395 (2012), 78-85.
[28] R. Seneta; Regular Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag,
Berlin, 1976.
[29] S. Taliaferro; A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), 897904.
[30] H. Usami; On a singular elliptic boundary value problem in a ball, Nonlinear Anal. 13 (1989),
1163-1170.
[31] Z. Zhao; Positive solutions of nonlinear second order ordinary differential equations, Proc.
Amer. Math. Soc. 121 (1994), 465-469.
Habib Mâagli
Department of Mathematics, College of Sciences and Arts, King Abdulaziz University,
Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia
E-mail address: habib.maagli@fst.rnu.tn
Ramzi Alsaedi
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box
80203, Jeddah 21589, Saudi Arabia
E-mail address: ramzialsaedi@yahoo.co.uk
Noureddine Zeddini
Department of Mathematics, College of Sciences and Arts, King Abdulaziz University,
Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia
E-mail address: noureddine.zeddini@ipein.rnu.tn