1. No category

Opuscula Math. 35, no. 1 (2015), 5–19
http://dx.doi.org/10.7494/OpMath.2015.35.1.5
Opuscula Mathematica
POSITIVE SOLUTIONS
WITH SPECIFIC ASYMPTOTIC BEHAVIOR
FOR A POLYHARMONIC PROBLEM ON Rn
Abdelwaheb Dhifli
Communicated by Vicentiu D. Radulescu
Abstract. This paper is concerned with positive solutions of the semilinear polyharmonic
equation (−∆)m u = a(x)uα on Rn , where m and n are positive integers with n > 2m,
α ∈ (−1, 1). The coefficient a is assumed to satisfy
a(x) ≈ (1 + |x|)−λ L(1 + |x|)
x ∈ Rn ,
for
0
(t)
−→ 0 as t −→ ∞; if λ = 2m,
where λ ∈ [2m, ∞) and L ∈ C 1 ([1, ∞)) is positive with tL
L(t)
R ∞ −1
one also assumes that 1 t L(t)dt < ∞. We prove the existence of a positive solution u
such that
e
e + |x|) for x ∈ Rn ,
u(x) ≈ (1 + |x|)−λ L(1
e := min(n − 2m, λ−2m ) and a function L,
e given explicitly in terms of L and satisfying
with λ
1−α
the same condition at infinity. (Given positive functions f and g on Rn , f ≈ g means that
c−1 g ≤ f ≤ cg for some constant c > 1.)
Keywords: asymptotic behavior, Dirichlet problem, Schauder fixed point theorem, positive
bounded solutions.
Mathematics Subject Classification: 34B18, 35B40, 35J40.
1. INTRODUCTION
This paper is concerned with positive solutions of the semilinear polyharmonic equation
(−∆)m u = a(x)uα on Rn (n ≥ 3) (in the sense of distributions),
(1.1)
where m and n are positive integers with n > 2m, α ∈ (−1, 1). The coefficient a is a
positive measurable function on Rn assumed to satisfy
−λ
a(x) ≈ (1 + |x|)
L(1 + |x|) for x ∈ Rn ,
c AGH University of Science and Technology Press, Krakow 2015
5
6
Abdelwaheb Dhifli
0
(t)
where λ ∈ [2m, ∞) and L ∈ C 1 ([1, ∞)) is positive with tL
L(t) −→ 0 as t −→ ∞; if
R ∞ −1
λ = 2m, one also assumes that 1 t L(t)dt < ∞. Here and throughout the paper,
for positive functions f and g on a set S, the notation f ≈ g means that there exists
a constant c > 1 such that c−1 g ≤ f ≤ cg on S.
Recently, by applying Karamata regular variation theory, the authors in [7], studied equation (1.1) in the unit ball of Rn (n ≥ 2) with Dirichlet boundary conditions.
They proved the existence of a continuous solution and gave an asymptotic behavior
of such a solution.
For the case m = 1 and α < 1, the pure elliptic equation
−∆u = a(x)uα ,
x ∈ Ω ⊂ Rn ,
(1.2)
has been extensively studied for both bounded and unbounded domain Ω in Rn
(n ≥ 2). We refer to [1, 2, 4–6, 8, 10–16, 18, 21] and the references therein, for various existence and uniqueness results related to solutions for the above equation
with homogeneous Dirichlet boundary conditions. In particular, many authors studied the exact asymptotic behavior of solutions of equation (1.2), see for example
[4–6, 10, 12, 13, 15, 16, 21]. For instance in [4], the authors studied (1.2) in Rn (n ≥ 3).
Thanks to the sub and supersolution method, they showed that equation (1.2) has
a unique positive classical solution which satisfies homogeneous Dirichlet boundary
conditions. Moreover, by applying Karamata regular variation theory, they improved
and extended the estimates established in [2,8,14]. On the other hand when α = 0 and
the equation (1.2) involves a degenerate operator p-Laplacian, Cavalheiro in [3] proved
the existence and uniqueness solution under a suitable condition on the function a.
Also, the result of equation (1.2) is extended to a class of elliptic systems. We refer
for example to [9] and [20] where the authors proved the existence and asymptotic
behavior of continuous solutions.
In this work, we generalize the results of [4] to equation (1.1). Note that the sub
and supersolution method is not available for (1.1). Then, we have to work around
this obstacle and we shall use the Schauder fixed-point theorem which requires invariance of a convex set under a suitable integral operator. Hence, we are restricted to
considering only the case α ∈ (−1, 1).
To simplify our statements, we refer to B + (Rn ) the set of Borel measurable nonnegative functions in Rn and C0 (Rn ) the class of continuous functions in Rn vanishing
continuously at infinity. Also, we use K to denote the set of functions L defined on
[1, ∞) by
!
Zt
z(s)
ds ,
L(t) := c exp
s
1
where z ∈ C([1, ∞)) such that limt−→∞ z(t) = 0 and c > 0.
For the rest of the paper, we use the letter c to denote a generic positive constant
which may vary from line to line.
Remark 1.1. It is obvious to see that L ∈ K if and only if L is a positive function
0
(t)
in C 1 [1, ∞) such that limt→∞ tL
L(t) = 0.
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
7
Example 1.2. Let p ∈ N∗ , (λ1 , λ2 , . . . , λp ) ∈ Rp and ω be a positive real number
sufficiently large such that the function
L(t) =
p
Y
(logk (ωt))−λk
k=1
is defined and positive on [1, ∞), where logk (x) = (log ◦ log ◦ . . . ◦ log)(x) (k times).
Then L ∈ K.
Throughout this paper, we denote by Gk,n (x, y) = Ck,n |x−y|1n−2k , the Green function of the operator (−∆)k in Rn , where Ck,n =
Γ( n
2 −k)
n
4k π 2 (k−1)!
,1≤k≤m<
n
2.
The function Gk,n (x, y) satisfies, for 2 ≤ k ≤ m,
Z
Gk,n (x, y) = G1,n (x, z)Gk−1,n (z, y)dz.
Rn
We define the k-potential kernel Vk,n on B + (Rn ) by
Z
Vk,n f (x) = Gk,n (x, y)f (y)dy.
Rn
Hence, for any f ∈ B + (Rn ) such that f ∈ L1loc (Rn ) and Vk,n f ∈ L1loc (Rn ), then we
have (−∆)k Vk,n f = f in the sense of distributions.
Now we are ready to present our main result.
Theorem 1.3. Equation (1.1) has a positive and continuous solution u satisfying for
x ∈ Rn
u(x) ≈ θ(x),
where the function θ is defined on Rn by
 
1
 1−α


Z∞


L(t) 



dt



t



 |x|+1



1
 (1 + |x|) 2m−λ
1−α
(L(1 + |x|)) 1−α
θ (x) :=
1

 1−α


2+|x|

Z


L(t) 

2m−n 

(1 + |x|)
dt




t


1




2m−n
(1 + |x|)
if λ = 2m,
if 2m < λ < n − (n − 2m)α,
(1.3)
if λ = n − (n − 2m)α,
if λ > n − (n − 2m)α.
Our idea in Theorem 1.3 above is based on the Schauder fixed-point method and
the convex set invariant under the integral operators. We note that (1.1) is formally
equivalent to the integral equation
u = Vm,n (auα ),
(1.4)
8
Abdelwaheb Dhifli
and u = Vm,n (a) is a solution in the linear case with α = 0. The asymptotic behavior
of Vm,n (a) is similar to that of a itself, only with different λ and L (Proposition 2.4).
Based on this observation, one constructs an asymptotic solution of (1.4), that is, a
function θ, again with the same kind of asymptotic behavior, such that θ ≈ Vm,n (aθα )
(Proposition 2.5). One then finds a constant c > 1 such that the operator u −→
Vm,n (auα ) maps the order interval [c−1 θ, cθ] ⊂ C0 (Rn ) into itself. The operator being
compact, Schauder’s fixed-point theorem yields a solution u of (1.4) with u ≈ θ,
proving our main result. We also control the asymptotic behavior of the iterated
Laplacians of u; in particular, u satisfies Navier boundary conditions at infinity.
The outline of the paper is as follows. In Section 2, we state some already known
results on functions in K, useful for our study and we give estimatises on some potential functions. Section 3 is reserved to the proof of our main result.
2. ESTIMATES AND PROPERTIES OF K
2.1. TECHNICAL LEMMAS
In what follows, we collect some fundamental properties of functions belonging to the
class K.
First, we need the following elementary result.
Lemma 2.1 ([19, Chap. 2, pp. 86–87]). Let γ ∈ R and L be a function in K. Then
we have:
R∞
R∞
γ+1
(i) If γ < −1, then 1 sγ L(s)ds converges and t sγ L(s)ds ∼ −t γ+1L(t) as
t −→ ∞,
R∞
Rt
γ+1
L(t)
as t −→ ∞.
(ii) If γ > −1, then 1 sγ L(s)ds diverges and 1 sγ L(s)ds ∼ t γ+1
Lemma 2.2 ([4]). (i) Let L1 , L2 ∈ K, p ∈ R. Then L1 L2 ∈ K and Lp1 ∈ K.
(ii) Let L ∈ K and η > 0. Then we have
L(t) ≈ L(t + η)
t ≥ 1,
for
lim t−η L(t) = 0,
(2.1)
lim tη L(t) = ∞.
(2.2)
t−→+∞
t−→+∞
(iii) Let L ∈ K. Then limt−→∞
L(t)
Rt
1
L(s)
s ds
= 0.
In particular, the function
Z1+t
t −→
L(s)
ds
s
is in K.
(2.3)
1
Further, if
R∞
1
L(s)
s ds
converges, then limt−→∞
L(t)
R ∞ L(s)
t
s
ds
= 0.
9
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
In particular, the function
Z∞
t −→
L(s)
ds
s
is in K.
(2.4)
t
The following behavior of the potential of radial functions on Rn is due to [17].
Lemma 2.3 ([17]). Let 0R ≤ j ≤ m − 1 and let f be a nonnegative radial measurable
∞
function in Rn such that 1 r2(m−j)−1 f (r)dr < ∞, then for x ∈ Rn , we have
Z∞
Vm−j,n f (x) ≈
rn−1
f (r)dr.
max(|x|, r)n−2(m−j)
(2.5)
0
2.2. ASYMPTOTIC BEHAVIOR OF SOME POTENTIAL FUNCTIONS
In what follows, we are going to give estimates on the potentials Vm−j,n a and
Vm−j,n (aθα ), for 0 ≤ j ≤ m − 1, where the function θ is given in (1.3).
Proposition 2.4. For 0 ≤ j ≤ m − 1 and x ∈ Rn
Vm−j,n a(x) ≈ ψ(|x|),
where ψ is the function defined on [0, ∞) by
 ∞
Z


L(r)


dr
if λ = 2(m − j),


r



t+1



 (1 + t)2(m−j)−λ L(1 + t)
if 2(m − j) < λ < n,
ψ (t) =
2+t
Z



L(r)

2(m−j)−n

dr if λ = n,
(1
+
t)


r



1



(1 + t)2(m−j)−n
if λ > n.
R ∞ 2(m−j)−1−λ
Proof. Let λ ≥ 2(m − j) and L ∈ K satisfying 1 t
L(t)dt < ∞ and such
that
L(1 + |x|)
.
a(x) ≈
(1 + |x|)λ
Thus, by (2.5), we have
Z∞
Vm−j,n a(x) ≈
rn−1
L(1 + r)
dr := I(|x|),
n−2(m−j)
(1 + r)λ
max(|x|, r)
0
where I is the function defined on [0, ∞) by
Z∞
I(t) =
0
rn−1
L(1 + r)
dr.
max(t, r)n−2(m−j) (1 + r)λ
10
Abdelwaheb Dhifli
So to prove the result, it is sufficient to show that I(t) ≈ ψ(t) for t ∈ [0, ∞).
Let t ≥ 1. We have
I(t) =
1
Z1
tn−2(m−j)
1
rn−1 L(1 + r)
dr + n−2(m−j)
(1 + r)λ
t
0
Z∞
+
Zt
rn−1 L(1 + r)
dr
(1 + r)λ
1
r2(m−j)−1 L(1 + r)
dr
(1 + r)λ
t
≈
1
Z1
tn−2(m−j)
rn−1 L(1 + r)
1
dr + n−2(m−j)
(1 + r)λ
t
0
Z∞
+
Zt
rn−1−λ L(r)dr
1
r2(m−j)−1−λ L(r)dr
t
:= I1 (t) + I2 (t) + I3 (t).
It is clear that
1
.
tn−2(m−j)
To estimate I2 and I3 , we distinguish two cases.
Case 1. λ > 2(m − j). Using Lemma 2.1 (i), we have for t ≥ 1
I1 (t) ≈
I3 (t) ≈
L(t)
.
tλ−2(m−j)
(2.6)
(2.7)
R∞
If 2(m−j) < λ < n, then applying again Lemma 2.1 (ii), we have 1 rn−1−λ L(r)dr =
Rt
∞ and 1 rn−1−λ L(r)dr ∼ tn−λ L(t), as t −→ ∞. So for t ≥ 1 we obtain
I2 (t) ≈
L(t)
.
tλ−2(m−j)
(2.8)
Then, by (2.6), (2.7), (2.8) and (2.1), for t ≥ 1 we have
I(t) ≈
1
L(t)
L(t)
+ λ−2(m−j) ≈ λ−2(m−j) .
tn−2(m−j)
t
t
Now, since the function t −→ I(t) and t −→
in [0, ∞), we obtain for t ≥ 0
L(1+t)
(1+t)λ−2(m−j)
are positive and continuous
L(1 + t)
.
(1 + t)λ−2(m−j)
R∞
If λ > n, then using Lemma 2.1 (i), we have 1 rn−1−λ L(r)dr < ∞ and for t ≥ 2,
I(t) ≈
Zt
1
rn−1−λ L(r)dr ≈ 1.
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
11
So we obtain, for t ≥ 1
I2 (t) ≈ t2(m−j)−n .
Moreover, using (2.1), we have for t ≥ 1, I1 (t) + I2 (t) + I3 (t) ≈ t2(m−j)−n (2 + tL(t)
λ−n ) ≈
t2(m−j)−n .
Then we deduce that for t ≥ 0,
I(t) ≈ (1 + t)2(m−j)−n .
Zt
1
If λ = n, we have I2 (t) ≈
tn−2(m−j)
L(r)
dr and I3 (t) ≈
r
L(t)
,
tn−2(m−j)
then using (2.6)
1
and (2.3), for t ≥ 1, we have
I(t) ≈
Zt
1
tn−2(m−j)
1+
L(r)
1
dr + L(t) ≈ n−2(m−j)
r
t
1
Zt
L(r)
dr.
r
1
So we obtain for t ≥ 0
I(t) ≈
Z2+t
1
(1 +
t)n−2(m−j)
L(r)
dr.
r
1
Z∞
Case 2. λ = 2(m − j). By Lemma 2.1 (ii),
rn−1−2(m−j) L(r)dr = ∞ and for t ≥ 2,
1
Zt
rn−1−2(m−j) L(r)dr ≈ tn−2(m−j) L(t).
1
Then we have for t ≥ 1, I2 (t) ≈ L(t). So for t ≥ 1, we have
I(t) ≈
Z∞
1
tn−2(m−j)
+ L(t) +
t
Hence using (2.2) and (2.4), for t ≥ 1, we have
Z∞
I(t) ≈
L(r)
dr.
r
t
So for t ≥ 0, we obtain
Z∞
I(t) ≈
t+1
This completes the proof.
L(r)
dr.
r
L(r)
dr.
r
12
Abdelwaheb Dhifli
The following proposition plays a crucial role in this paper.
Proposition 2.5. Let θ be the function given in (1.3). Then we have for x ∈ Rn and
for 0 ≤ j ≤ m − 1
e
Vm−j,n (aθα )(x) ≈ θ(x),
where θe is the function defined on Rn by

1
 1−α

∞
Z



L(t) 


if λ = 2m and j = 0,
dt


t


 |x|+1



α

 1−α


∞

Z


L(t) 


−2j


if λ = 2m and 0 < j ≤ m − 1,
dt
(1 + |x|) L(1 + |x|) 

t
|x|+1
θe (x) :=

λ−2(m−j)−2αj
1

−

1−α

(1 + |x|)
(L(1 + |x|)) 1−α
if 2m < λ < n − (n − 2m)α,



1




1−α

2+|x|

Z


L(t)




dt
if λ = n − (n − 2m)α,
(1 + |x|)−(n−2(m−j)) 


t



1




(1 + |x|)−(n−2(m−j))
if λ > n − (n − 2m)α.
R ∞ 2m−1−λ
Proof. Let λ ≥ 2m and L ∈ K satisfying 1 t
L(t)dt < ∞ and such that
a(x) ≈ (1 + |x|)−λ L(1 + |x|).
Then for every x ∈ Rn , we have
a(x)θα (x)

α

 1−α

∞
Z



L(t) 



(1 + |x|)−λ L(1 + |x|) 
dt


t



|x|+1



1

− λ−2mα

(1 + |x|) 1−α (L(1 + |x|)) 1−α
≈ h(x) :=
α

 1−α

2+|x|

Z



L(t) 



dt
(1 + |x|)−n L(1 + |x|) 


t



1




(1 + |x|)−(λ−(n−2m)α) L(1 + |x|)
if λ = 2m,
if 2m < λ < n − (n − 2m)α,
if λ = n − (n − 2m)α,
if λ > n − (n − 2m)α.
So, one can see that
e + |x|),
h(x) := (1 + |x|)−µ L(1
where µ ≥ 2(m − j) for all 0 ≤ j ≤ m − 1 and by Lemma 2.2 (i) and (iii), we have
e ∈ K.
L
e and λ by µ.
The result follows from Proposition 2.4 by replacing L by L
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
13
Remark 2.6.
(i) Let θ be the function given in (1.3). Then from Proposition 2.5 for j = 0 and
x ∈ Rn , we have
Vm,n (aθα )(x) ≈ θ(x).
(ii) We mention that if α = 0 and the function a satisfies an exact asymptotic
behavior, the previous results in i) have been stated in Proposition 2.4.
Remark 2.7. By using (2.5) and Proposition 2.5, we see that for 0 ≤ j ≤ m − 1
Z∞ α
θ (r)L(1 + r)
dr < ∞.
rλ−2(m−j)+1
1
Proposition 2.8. Let 0 ≤ j ≤ m − 1 and let θ the function given in (1.3). Then the
family of functions
Z
a(y)uα (y)
Λ = x −→ T u(x) :=
dy;
|x − y|n−2(m−j)
n
R
1
θ ≤ u ≤ Cθ with C > 0 is a fixed constant
C
is uniformly bounded and equicontinuous in C0 (Rn ). Consequently, Λ is relatively
compact in C(Rn ∪ {∞}).
Proof. Let 0 ≤ j ≤ m − 1, x0 ∈ Rn , R > 0 and let u be a positive function satisfying
1
θ ≤ u ≤ Cθ.
C
For x, x0 ∈ B(x0 , R), we have
|T u(x) − T u(x0 )|
Z
a(y)θα (y)
dy +
≤c
|x − y|n−2(m−j)
|x−y|≤3R
Z
+
Z
|x0
a(y)θα (y)
dy
− y|n−2(m−j)
|x0 −y|≤5R
!
2(m−j)−n
0
2(m−j)−n
α
|x − y|
a(y)θ (y)dy
− |x − y|
|x−y|≥3R
|x|+3R
Z
≤c
r2(m−j)−1 L(1 + r)θα (r)
dr
(1 + r)λ
(|x|−3R)+
0
|xZ
|+5R
+
r2(m−j)−1 L(1 + r)θα (r)
dr
(1 + r)λ
(|x0 |−5R)+
Z
+
|x−y|≥3R
!
α
L(1
+
|y|)θ
(|y|)
x − y|2(m−j)−n − |x0 − y|2(m−j)−n dy .
(1 + |y|)λ
14
Abdelwaheb Dhifli
We deduce from Remark 2.7, for 0 ≤ j ≤ m − 1, that the function
Zt
ϕ(t) :=
r2(m−j)−1
L(1 + r)θα (r)dr
(1 + r)λ
0
is continuous in [0, ∞). This implies that
|x|+3R
Z
r2(m−j)−1 L(1 + r)θα (r)
dr = ϕ(|x| + 3R) − ϕ((|x| − 3R)+ ) −→ 0 as R −→ 0.
(1 + r)λ
(|x|−3R)+
As in the above argument, we get
0
|xZ
|+5R
lim
R−→0
(|x0 |−5R)+
r2(m−j)−1 L(1 + r)θα (r)
dr = 0.
(1 + r)λ
If |x − y| ≥ 3R and since |x − x0 | < 2R, then |x0 − y| ≥ R and hence
|x − y|2(m−j)−n − |x0 − y|2(m−j)−n ≤ (32(m−j)−n + 1)R2(m−j)−n .
We deduce by the dominated convergence theorem and Remark 2.7 that
Z
α
|x − y|2(m−j)−n − |x0 − y|2(m−j)−n L(1 + |y|)θ (y) dy −→ 0 as |x − x0 | −→ 0.
(1 + |y|)λ
|x−y|≥3R
It follows that
|T u(x) − T u(x0 )| −→ 0 as |x − x0 | −→ 0,
which implies that T u is continuous in Rn . Moreover, since
e
T u(x) ≈ Vm−j,n (aθα )(x) ≈ θ(x),
where θe is the function, given in Proposition 2.5.
e
From Lemma 2.2, we deduce that θ(x)
−→ 0 as |x| −→ ∞. Then Λ is equicontinuous in Rn . Moreover, the family {T u(x); C1 θ ≤ u ≤ Cθ} is uniformly bounded in Rn .
It follows from Ascoli’s theorem that Λ is relatively compact in C(Rn ∪ {∞}).
3. PROOF OF THEOREM 1.3
Proof of Theorem 1.3. The aim of this section is to prove the existence of a positive
solution of equation (1.1) and to give the asymptotic behavior of such a solution.
Our idea is based on the Schauder fixed-point method and the convex set invariant
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
15
under the integral operators. By Remark 2.6 (i), there exists c0 > 0 such that for each
x ∈ Rn
1
θ(x) ≤ Vm,n (aθα )(x) ≤ c0 θ(x).
c0
Let c > 1 such that c1−|α| ≥ c0 . In order to apply a fixed point argument, we consider
the following convex set given by
1
n
Y := u ∈ C0 (R ); θ ≤ u ≤ cθ .
c
Then Y is a nonempty closed bounded in C0 (Rn ). Let T be the integral operator
defined on Y by
Z
a(y)uα (y)
α
dy, x ∈ Rn .
T u(x) := Vm,n (au )(x) = Cm,n
|x − y|n−2m
Rn
Since for every u ∈ Y and −1 < α < 1, c−|α| θα ≤ uα ≤ c|α| θα , then we get
1
1
θ ≤ c−|α| θ ≤ c−|α| Vm,n (aθα ) ≤ T u ≤ c|α| Vm,n (aθα ) ≤ c|α| c0 θ ≤ cθ.
c
c0
Thus T Y ⊂ Y and we conclude by Proposition 2.8 that T Y is relatively compact in
C(Rn ∪ {∞}).
Next, let us prove the continuity of T in the uniform norm. Let (uk ) be a sequence
in Y which converges uniformly to u ∈ Y and let x ∈ Rn , we have
Z
α
|T uk (x) − T u(x)| ≤ c |x − y|2m−n a(y)|uα
k (y) − u (y)|dy
Rn
and
α
α
|uα
k (y) − u (y)| ≤ cθ (y).
Then, we deduce by the dominated convergence theorem, for x ∈ Rn , T uk (x) −→
T u(x) as k −→ ∞. Finally, since T Y is a relatively compact family in C(Rn ∪ {∞}),
then
kT uk − T uk∞ −→ 0 as k −→ ∞.
We have proved that T is a compact mapping from Y to itself. So the Schauder fixed
point theorem implies the existence of u ∈ Y which satisfies the integral equation
Z
a(y)uα (y)
u(x) = Cm,n
dy = Vm,n (auα )(x).
|x − y|n−2m
Rn
Hence, applying (−∆)m on both sides of the equation above, we obtain in the sense
of distributions
(−∆)m u = auα .
16
Abdelwaheb Dhifli
Moreover, by iterated Green function, we have in the sense of distributions, for
0 ≤ j ≤ m − 1,
(−∆)j u = Vm−j,n (auα ).
It follows from Proposition 2.5 that
e
Vm−j,n (auα )(x) ≈ Vm−j,n (aθα )(x) ≈ θ(x),
where θe is the function given in Proposition 2.5. Moreover, using (2.1) we have
e
θ(x)
−→ 0 as |x| −→ ∞.
This ends the proof.
We end this section by some examples and remarks.
Example 3.1. Let a be a nonnegative function in Rn satisfying for x ∈ Rn ,
a(x) ≈ (1 + |x|)−λ (log ω(1 + |x|))−µ ,
where λ ≥ 2m, µ > 1 and ω is a positive constant large enough.
Then using Theorem 1.3, equation (1.1) has a positive continuous solution u in
Rn satisfying the following estimates:
(i) If λ = 2m, then for x ∈ Rn
u(x) ≈ log ω(1 + |x|)
1−µ
1−α
.
(ii) If 2m < λ < n − α(n − 2m), then for x ∈ Rn
u(x) ≈ (1 + |x|)−
λ−2m
1−α
−µ
log ω(1 + |x|) 1−α .
(iii) If λ ≥ n − α(n − 2m), then for x ∈ Rn
u(x) ≈ (1 + |x|)2m−n .
Remark 3.2. It is clear from Theorem 1.3 that the solution of equation (1.1) satisfies
for x ∈ Rn
λ−2m
u(x) ≈ (1 + |x|)− min(n−2m, 1−α ) ψL,α,λ,m (1 + |x|),
where ψL,α,λ,m is a function defined in [1, ∞) by
 
1
 1−α

Z∞




 L(s) ds
if λ = 2m,



s


t



1
 (L(t)) 1−α
if 2m < λ < n − (n − 2m)α,
ψL,α,λ,m (t) :=
1
 1+t
 1−α


Z


L(s) 



ds
if λ = n − (n − 2m)α,



s


1



1
if λ > n − (n − 2m)α.
We conclude from Lemma 2.2 that the function ψL,α,λ,m is in K.
(3.1)
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
17
Example 3.3. Let a1 and a2 be positive functions in Rn such that
a1 (x) ≈ (1 + |x|)−λ1 L1 (1 + |x|),
and
a2 (x) ≈ (1 + |x|)−λ2 L2 (1 + |x|),
where for i ∈ {1, 2}, λi ∈ R and Li ∈ K. We consider the following system:
11
(−∆)m1 u1 = a1 (x)uα
1 ,
α12 α22
m2
(−∆) u2 = a2 (x)u1 u2 ,
(3.2)
where n > 2 max(m1 , m2 ), α11 , α22 ∈ (−1, 1) and αR12 ∈ R. We suppose that λ1 ∈
∞
[2m1 , ∞) and if λ1 = 2m1 , one also assumes that 1 t−1 L1 (t)dt < ∞. By Theorem 1.3, there exists a positive continuous solution u1 to the equation
11
(−∆)m1 u1 = a1 (x)uα
1 .
Besides u1 satisfies for x ∈ Rn
u1 (x) ≈ (1 + |x|)−γ ψL1 ,α11 ,λ1 ,m1 (1 + |x|),
(3.3)
1 −2m1
where γ := min(n − 2m1 , λ1−α
) and ψL1 ,α11 ,λ1 ,m1 is the function defined in (3.1)
11
by replacing L by L1 , α by α11 , λ by λ1 and m by m1 .
Now, suppose that λ2 + γ α12 ∈ [2m2 , ∞) and if λ2 + γ α12 = 2m2 , one also assumes
that
Z∞
α12
(t)dt < ∞.
t−1 L2 (t)ψL
1 ,α11 ,λ1 ,m1
1
Applying again Theorem 1.3, we deduce that equation
12 α22
(−∆)m2 u2 = a2 (x)uα
1 u2
has a positive continuous solution u2 which satisfies for x ∈ Rn
u2 (x) ≈ (1 + |x|)− min(n−2m2 ,
λ2 +γ α12 −2m2
1−α22
)
ψLf2 ,α22 ,λ,m2 (1 + |x|),
(3.4)
α12
f2 := L2 ψ α12
where L
. Hence, the system (3.2) has positive
L1 ,α11 ,λ1 ,m1 and λ := λ2 + γ
continuous solutions u1 , u2 which satisfy (3.3) and (3.4), respectively.
Remark 3.4. We note that in the example above due to Remark 3.2 and Lemma 2.2,
Theorem 1.3 can be applied recursively to systems of the form
(−∆)mk uk = ak (x)
k
Y
j=1
k−1
Y α α
α
uj jk = ak (x)
uj jk uk kk ,
j=1
∗
k ∈ {1, 2, . . . , K}, K ∈ N , under suitable assumptions on the coefficient and exponents.
Acknowledgments
The author thanks Professor Habib Mâagli for useful suggestions.
The author also thanks the referee for his/her careful reading of the paper.
18
Abdelwaheb Dhifli
REFERENCES
[1] S. Ben Othman, H. Mâagli, S. Masmoudi, M. Zribi, Exact asymptotic behavior near the
boundary to the solution for singular nonlinear Dirichlet problems, Nonlinear Anal. 71
(2009), 4137–4150.
[2] H. Brezis, S. Kamin, Sublinear elliptic equations in Rn , Manuscripta Math. 74 (1992),
87–106.
[3] A.C. Cavalheiro, Existence results for Dirichlet problems with degenerated p-Laplacian,
Opuscula Math. 33 (2013) 3, 439–453.
[4] R. Chemmam, A. Dhifli, H. Mâagli, Asymptotic bhavior of ground state solutions for
sublinear and singular nonlinear Dirichlet problems, Electron. J. Differential Equations
2011 (2011) 88, 1–12.
[5] R. Chemmam, H. Mâagli, S. Masmoudi, M. Zribi, Combined effects in nonlinear singular
elliptic problems in a bounded domain, Adv. Nonlinear Anal. 1 (2012) 4, 391–404.
[6] M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular
nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222.
[7] A. Dhifli, Z. Zine El Abidine, Asymptotic behavior of positive solutions of a semilinear
polyharmonic problem in the unit ball, Nonlinear Anal. 75 (2012), 625–636.
[8] A.L. Edelson, Entire solutions of singular elliptic equations, J. Math. Anal. 139 (1989),
523–532.
[9] A. Ghanmi, H. Mâagli, V. Rˇ
adulescu, N. Zeddini, Large and bounded solutions for a
class of nonlinear Schrödinger stationary systems, Anal. Appl. (Singap.) 7 (2009) 4,
391–404.
[10] M. Ghergu, V.D. Radulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler
equation, C.R. Acad. Sci. Paris. Ser. I 337 (2003), 259–264.
[11] M. Ghergu, V.D. Radulescu, Sublinear singular elliptic problems with two parameters,
J. Differential Equations 195 (2003), 520–536.
[12] M. Ghergu, V.D. Radulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl. 333 (2007),
265–273
[13] S. Gontara, H. Mâagli, S. Masmoudi, S. Turki, Asymptotic behavior of positive solutions
of a singular nonlinear Dirichlet, J. Math. Anal. Appl. 369 (2010), 719–729.
[14] A.V. Lair, A.W. Shaker, Classical and weak solutions of a singular semilinear elliptic
problem, J. Math. Anal. Appl. 211 (1997), 371–385.
[15] A.C. Lazer, P.J. Mckenna, On a singular nonlinear elliptic bondary-value problem, Proc.
Amer. Math. Soc 111 (1991), 721–730.
[16] H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem,
Nonlinear Anal. 74 (2011), 2941–2947.
[17] H. Mâagli, M. Zribi, Existence of positive solutions for some polyharmonic nonlinear
equations in Rn , Abstr. Appl. Anal. 2006 (2005), 1–24.
Positive solutions with specific asymptotic behavior for a polyharmonic problem. . .
19
[18] C.A. Santos, On ground state solutions for singular and semilinear problems including
super linear terms at infinity, Nonlinear Anal. 71 (2009), 6038–6043.
[19] R. Seneta, Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, Berlin, 1976.
[20] S. Turki, Existence and asymptotic behavior of positive continuous solutions for a nonlinear elliptic system in the half space, Opuscula Math. 32 (2012) 4, 783–795
[21] Z. Zhang, The asymptotic behaviour of the unique solution for the singular Lane-Emdem-Fowler equation, J. Math. Anal. Appl. 312 (2005), 33–43.
Abdelwaheb Dhifli
[email protected]
Campus Universitaire
Faculté des Sciences de Tunis
Département de Mathématiques
2092 Tunis, Tunisia
Received: March 3, 2014.
Revised: April 4, 2014.
Accepted: April 24, 2014.