Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 239, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
POSITIVE SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS
ADEL BEN DEKHIL
Abstract. In this article, we study the existence of positive solutions for the
system
∆u = H(x, u, v),
∆v = K(x, u, v), in Rn (n ≥ 3),
where H, K : Rn × [0, ∞) × [0, ∞) → [0, ∞) are continuous functions satisfying
H(x, u, v) ≤ p1 (|x|)F (u + v) and K(x, u, v) ≤ q1 (|x|)G(u + v). In terms of
the growth of the variable potential functions p1 , q1 and the nonlinearities F
and G, we establish some sufficient conditions for the existence of positive
continuous solutions for this system and we discuss whether these solutions
are bounded or blow up at infinity.
1. Introduction
Semilinear elliptic systems of the form
∆u = H(x, u, v),
in Rn (n ≥ 3),
∆v = K(x, u, v),
in Rn ,
(1.1)
have been studied intensively in the previous few years and various results concerning the existence and nonexistence of positive entire large or bounded solutions have
been obtained. We refer the reader to [2, 3, 4, 5, 6, 9, 11, 12, 15, 16, 18, 19, 20, 21]
and their references for recent results concerning the existence and qualitative analysis of solutions of (1.1).
The interest in systems of nonlinear stationary equations is motivated by applications to theory of Newtonian fluids and nonlinear optics. More precisely, coupled
nonlinear stationary systems arise in the description of several physical phenomena such as the propagation of pulses in birefringent optical fibers and Kerr-like
photorefractive media, see [1, 13].
When H(x, u, v) = p(|x|)v α , K(x, u, v) = q(|x|)uβ , 0 < α ≤ β, Lair and Wood
in [11] considered the existence and nonexistence of entire positive radial solutions
to (1.1) under the conditions of integrability or non integrability of the functions
r → rp(r) and r → rq(r) on (0, ∞). Their results were extended by Cı̂rstea and
Rădulescu [3], Wang and Wood [18], Ghergu and Rădulescu [6], Peng and Song [15],
Ghanmi et al [5], Li et al [12], Zhang [20]. In [21], the authors considered the case
2000 Mathematics Subject Classification. 35B08, 35B09, 35J47.
Key words and phrases. Semilinear elliptic systems; positive large solution; positive bounded
solution.
c
2012
Texas State University - San Marcos.
Submitted October 14, 2012. Published December 28, 2012.
1
2
A. BEN DEKHIL
EJDE-2012/239
where H(x, u, v) = p(x)f (v), K(x, u, v) = q(x)g(u) with p, q nontrivial nonnegative
continuous on [0, ∞) and f, g continuous , nondecreasing on [0, ∞) that are positive
on (0, ∞). In the case where p, q are radial and under the condition
Z ∞Z t
−1/2
(f (s) + g(s))ds
dt = ∞ ,
(1.2)
1
0
they proved that (1.1) has one positive solution (u, v). Moreover, if
Z ∞
Z ∞
sp(s) ds =
sq(s) ds = ∞ ,
0
0
then every positive radial entire solution (u, v) of (1.1) is large (i.e. limx→∞ u(x) =
limx→∞ v(x) = ∞). And if
Z ∞
Z ∞
sp(s) ds < ∞ and
sq(s) ds < ∞ ,
0
0
then every positive radial entire solution (u, v) of (1.1) is bounded.
By using the sub-supersolution method, they establish some conditions on p, q in
order to prove the existence of positive bounded solutions in the case where p, q are
non radial. Their results extend partially those of Zhang [20], where the existence
of entire positive
large solutions or bounded ones for (1.1) was studied under
R ∞radial
ds
= ∞.
the condition 1 f (s)+g(s)
Their results do not cover the cases where
H(x, u, v) = a1 (x)v α1 + b1 (x)uγ1 + c1 (x)(u + v)β1 + d1 (x)uδ1 v λ1 ,
K(x, u, v) = a2 (x)v α2 + b2 (x)uγ2 + c2 (x)(u + v)β2 + d2 (x)uδ2 v λ2 ,
where αi , βi , γi , δi , λi ∈ (0, ∞).
Our aim in this paper is to extend the results in [4] for the particular case of
the Dirichlet laplacian and to extend those in [21] to a wider class of functions
H(x, u, v) and K(x, u, v). More precisely, our results apply in particular to the
previous examples of functions H, K and to the case where H(x, u, v) = p(x)f (u, v)
, K(x, u, v) = q(x)g(u, v) with f and g are nondecreasing with respect to first and
the second variables. To this aim, we assume that H and K satisfy the following
hypotheses:
(H1) H, K : Rn × [0, ∞) × [0, ∞) → [0, ∞) are continuous.
(H2) There exist nonnegative functions pi , qi , fi , gi , 1 ≤ i ≤ 2, F and G satisfying
for each x ∈ Rn and (u, v) ∈ [0, ∞) × [0, ∞),
p2 (|x|)g1 (v)g2 (u) ≤ H(x, u, v) ≤ p1 (|x|)F (u + v),
q2 (|x|)f1 (u)f2 (v) ≤ K(x, u, v) ≤ q1 (|x|)G(u + v),
with fi , gi , 1 ≤ i ≤ 2, F , G : [0, ∞) → [0, ∞) are nondecreasing continuous,
positive on (0, ∞) and pi , qi : [0, ∞) → [0, ∞) are continuous.
(H3) There exist c > 0 such that
Z tp
Λ(s)ds < Lc (∞) := lim Lc (r),
0
r→∞
EJDE-2012/239
POSITIVE SOLUTIONS
for all t > 0, where for α > 0,
Z t
ds
p
, t ≥ α,
Lα (t) =
2(F(s) + G(s))
α
Λ(r) = max (p1 (s) + q1 (s)), r ≥ 0 ,
s∈[0,r]
Z r
Z r
F(r) =
F (s)ds , G(r) =
G(s)ds , r ≥ 0.
0
3
(1.3)
0
We note that Lα has an inverse function L−1
α from [α, ∞) to [0, Lα (∞)), where
Lα (∞) = limr→∞ Lα (r) ∈ (0, ∞].
Remark 1.1. If Lα (∞) = ∞, then
Z ∞
Z ∞
ds
ds
p
p
=
= ∞.
F(s)
G(s)
α
α
R∞
R∞
Remark 1.2. By [9], we see that if 1 √ds < ∞, then 1 Fds
(s) < ∞. In other
F (s)
R ∞ ds
R ∞ ds
R ∞ ds
words, if 1 F (s) = ∞, then 1 √
= ∞. Conversely, if 1 √
= ∞, then
F (s)
F (s)
R ∞ ds
= ∞ does not hold. For example, for β > 0 and F (t) = 2(1 + t)(ln(t +
1 F (s)
2β−1
1))
= (tR+ 1)2 (ln(t + 1))2β . So we can see that
R ∞ ds (ln(t + 1) + β), we have F(t)
∞
1
= ∞ if and only if 0 < β ≤ 2 and 1 √ds = ∞ if and only if 0 < β ≤ 1.
1 F (s)
F (s)
To discuss the existence of positive radial solutions to these nonlinear systems,
we study the system of nonlinear differential equations
1
(Ay 0 )0 = g(t, y, z), in (0, ∞),
A
1
(Bz 0 )0 = f (t, y, z), in (0, ∞),
B
lim+ A(t)y 0 (t) = lim+ B(t)z 0 (t) = 0,
t→0
(1.4)
t→0
y(0) = a > 0,
z(0) = b > 0,
where the continuous functions A, B : [0, ∞) → [0, ∞) are nondecreasing, differentiable and positive on (0, ∞). We assume that f and g satisfy the following
hypotheses.
(A1) f , g : (0, ∞) × [0, ∞) × [0, ∞) → [0, ∞) are continuous, nondecreasing with
respect to the second and the third variables.
(A2) There exist nonnegative functions hi , ki , ξi , ωi , 1 ≤ i ≤ 2, φ and ψ satisfying
for each (t, u, v) ∈ (0, ∞) × [0, ∞) × [0, ∞),
h2 (t)ω1 (v)ω2 (u) ≤ g(t, u, v) ≤ h1 (t)φ(u + v),
k2 (t)ξ1 (u)ξ2 (v) ≤ f (t, u, v) ≤ k1 (t)ψ(u + v),
with ξi , ωi , 1 ≤ i ≤ 2, φ, ψ : [0, ∞) → [0, ∞) are nondecreasing continuous,
positive on (0, ∞) and hi , ki : (0, ∞) → [0, ∞) continuous and satisfying
Z
Z 1
Z
Z 1
1 t
1 t
A(s) hi (s) ds dt < ∞,
B(s) ki (s) ds dt < ∞.
0 B(t)
0 A(t)
0
0
4
A. BEN DEKHIL
EJDE-2012/239
(A3) For all r > 0, we suppose that
√ Z rq
e
2
Λ(s)ds
< Ma+b (∞) ,
0
where, for α > 0,
Z
t
ds
, t≥α
2(Φ(s) + Ψ(s))
Z r
Z r
Φ(r) =
φ(s)ds, Ψ(r) =
ψ(s)ds , r ≥ 0
Mα (t) =
α
p
0
0
e
Λ(r)
= max (h1 (s) + k1 (s)),
r ≥ 0.
s∈[0,r]
For any nonnegative measurable functions ϕ in (0, ∞), we define
Z t
Z t
Z
Z
1 r
1 r
A(s)ϕ(s)ds dr , SB ϕ(t) =
B(s)ϕ(s)ds dr.
SA ϕ(t) =
0
0 B(r)
0
0 A(r)
Now, we are ready to give our existence result for (1.4).
Theorem 1.3. Under the hypotheses (A1)–(A3), system (1.4) has a positive solu2
tion (y, z) ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) satisfying for each t ∈ [0, ∞)
√ Z t q
−1
e
a + ω1 (b)ω2 (a)SA (h2 )(t) ≤ y(t) ≤ Ma+b
Λ(s)ds
,
2
0
√ Z t q
−1
e
b + ξ1 (a)ξ2 (b)SB (k2 )(t) ≤ z(t) ≤ Ma+b
2
Λ(s)ds
,
0
−1
Ma+b
where
[α, ∞).
is the inverse function of Ma+b which is defined from [0, Ma+b (∞)) to
Remark 1.4. In the case A(t) = B(t), the solution (y, z) of system (1.4) satisfies
Z tq
e
Λ(s)ds
,
a + ω1 (b)ω2 (a)SA (h2 )(t) ≤ y(t) ≤ M −1
a+b
0
Z tq
−1
e
Λ(s)ds
,
b + ξ1 (a)ξ2 (b)SA (k2 )(t) ≤ z(t) ≤ Ma+b
for t ≥ 0.
0
Now we give the existence result for system (1.1) under the following hypotheses.
(H4) The function r → r2(n−1) (p1 (r) + q1 (r)) is nondecreasing for large r.
(H5) There exists a positive constant ε such that
Z ∞
r1+ε (p1 (r) + q1 (r))dr < ∞,
0
Theorem 1.5. Under assumptions (H1)–(H5), system (1.1) has a positive entire
bounded continuous solution.
Example 1.6. Let αi , βi , γi , δi , λi ∈ (0, ∞) satisfying max(αi , βi , γi , δi + λi ) < 1
for 1 ≤ i ≤ 2 and 2 < σ < 2(n − 1). Put
H(x, u, v) = a1 (x)v α1 + b1 (x)uγ1 + c1 (x)(u + v)β1 + d1 (x)uδ1 v λ1 ,
K(x, u, v) = a2 (x)v α2 + b2 (x)uγ2 + c2 (x)(u + v)β2 + d2 (x)uδ2 v λ2 ,
EJDE-2012/239
POSITIVE SOLUTIONS
5
where and ai , bi , ci , di are nontrivial nonnegative continuous function in Rn satisfying
2 X
i=1
max ai (y) + max bi (y) + max ci (y) + max di (y) ≤
|y|≤s
|y|≤s
|y|≤s
|y|≤s
1
.
1 + sσ
Then problem (1.1) has a positive continuous bounded solution (u, v). Indeed, the
hypotheses (H1), (H2), (H4) and (H5) are clearly satisfied. Since max(αi , βi , γi , δi +
λi ) < 1 for 1 ≤ i ≤ 2 and
1
[(u + v)α1 + (u + v)β1 + (u + v)γ1 + (u + v)δ1 +λ1 ],
1 + |x|σ
1
K(x, u, v) ≤
[(u + v)α2 + (u + v)β2 + (u + v)γ2 + (u + v)δ2 +λ2 ]
1 + |x|σ
H(x, u, v) ≤
we deduce that L1 (∞) = ∞ and so hypothesis (H3) is satisfied.
Example 1.7. Let 2 < σ < 2(n − 1), and let β1 , β2 > 0 be such that β0 =
max(β1 , β2 ) > 1 and
1+
1
2
<
.
σ−2
2(β0 − 1)(Log(3))β0
(1.5)
Let p, q be two nontrivial nonnegative continuous functions in Rn such that
max p(y) + max q(y) ≤
|y|≤s
|y|≤s
1
.
1 + sσ
Then the problem
∆u = 2p(x)(u + v + 2)(Log(u + v + 2))2β1 −1 (Log(u + v + 2) + β1 )
∆v = 2q(x)(u + v + 2)(Log(u + v + 2))2β2 −1 (Log(u + v + 2) + β2 ) ,
in Rn with n ≥ 3, has a positive bounded continuous solution (u, v). Indeed, the
hypotheses (H1), (H2), (H4) and (H5) are clearly satisfied. Now, (H3) follows from
(1.5) and the fact that for each t ≥ 1,
Z 1
Z t
Z t
Z t
ds
ds
ds
ds
2
√
√
√
≤
+
≤
1
+
σ ≤ 1 +
σ
σ
σ
2
σ
−
2
s
1
+
s
1
+
s
1
+
s
0
0
1
1
and
Z ∞
1
Z
ds
p
2(s + 2)2 ((log(s + 2))2β1 + (log(s + 2))2β2 )
∞
ds
2(s
+
2)(Log(s
+ 2))β0
1
1
=
.
2(β0 − 1)(Log(3))β0 −1
≥
From Theorem 1.5, we have the following corollaries in the case where H(x, u, v) =
H(|x|, u, v) and K(x, u, v) = K(|x|, u, v).
Corollary 1.8. Under hypotheses (H1)–(H3), problem (1.1) has one positive solution. Under the additional hypothesis
R∞
R∞
(H6) 0 sp2 (s) ds = 0 sq2 (s) ds = ∞,
every positive radial entire solution (u, v) of (1.1) is large and satisfies
u(0) + g1 (v(0))g2 (u(0))P2 (r) ≤ u(r),
v(0) + f1 (u(0))f2 (v(0))Q2 (r) ≤ v(r) ,
6
A. BEN DEKHIL
for all r ≥ 0, where
Z r
Z t
1−n
P2 (r) =
t
sn−1 p2 (s) ds dt,
0
EJDE-2012/239
Z
r
Q2 (r) =
0
t
1−n
0
Z
t
sn−1 q2 (s) ds dt.
0
For the next corollary, we use the assumption
(H7) Lα (∞) = ∞.
Corollary 1.9. Assume that (H1), (H2), (H4), (H7) are satisfied. If (1.1) has a
nonnegative radial entire large solution, then
Z ∞
r1+ε (p1 (r) + q1 (r))dr = ∞, ∀ ε > 0.
0
An other result for the radial case is given under the assumption
√
(H8) p1 + q1 ∈ L1 (0, ∞).
Theorem 1.10. Assume that (H1), (H2), (H4), (H8) are satisfied. If (u, v) is
a positive entire large radial solution of (1.1), then F and G satisfy the KellerOsserman condition
Z ∞
ds
p
L1 (∞) =
< ∞.
2(F(s) + G(s))
1
2. Proof of main results
Proof of Theorem 1.3. Let (ym )m≥0 and (zm )m≥0 be the sequences of positive continuous functions defined on [0, ∞) by
Z
t
ym+1 (t) = a +
Z
0
t
zm+1 (t) = b +
0
y0 (t) = a, z0 (t) = b,
Z
1 r
A(s)g(s, ym (s), zm (s))ds dr,
A(r) 0
Z
1 r
B(s)f (s, ym (s), zm (s))ds dr.
B(r) 0
Clearly ym , zm ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) and are positive, so we deduce by (A1)
that (ym )m≥0 and (zm )m≥0 are nondecreasing sequences and for each m ∈ N, the
functions t → ym (t) and t → zm (t) are nondecreasing. Hence, for each t ∈ (0, ∞),
Z t
1
0
ym (t) =
A(s)g(s, ym−1 (s), zm−1 (s))ds,
A(t) 0
Z t
1
0
B(s)f (s, ym−1 (s), zm−1 (s))ds
zm
(t) =
B(t) 0
Which implies
0
0
(A(t)ym
(t)) = A(t)g(t, ym−1 (t), zm−1 (t)),
≤ A(t)g(t, ym (t), zm (t)).
0
2A(t)ym
(t)
Multiplying this expression by
and integrate on [0, t], we obtain
Z t
2
0
0
(A(t)ym
A2 (s)h1 (s) (φ(ym (s) + zm (s))) ym
(t)) ≤ 2
(s) ds
0
Z t
2
0
0
e
≤ 2Λ(t)A
(t)
(φ(ym (s) + zm (s))) (ym
(s) + zm
(s)) ds
0
2
e
= 2Λ(t)A
(t)
Z
ym (t)+zm (t)
φ(s) ds.
a+b
EJDE-2012/239
POSITIVE SOLUTIONS
7
Which implies
0
ym
(t) ≤
q
p
e
Λ(t)
2Φ(ym (t) + zm (t)).
Similarly, we have
0
zm
(t)
≤
q
p
e
Λ(t)
2Ψ(ym (t) + zm (t)).
Then
0
0
ym
(t) + zm
(t) ≤
p
√ q
e
2 Λ(t)
2(Φ(ym (t) + zm (t)) + Ψ(ym (t) + zm (t))).
Therefore,
Z
ym (t)+zm (t)
Ma+b (ym (t) + zm (t)) =
a+b
Z t
=
ds
p
2 (Φ(s) + Ψ(s))
ds
0
0
ym
(s) + zm
(s)
p
2 (Φ(ym (s) + zm (s)) + Ψ(ym (s) + zm (s)))
√ Z rq
e
≤ 2
Λ(s)ds.
0
ds
0
Since
−1
Ma+b
is increasing on [0, Ma+b (∞)), we obtain
√ Z t q
−1
e
2
Λ(s)ds
,
ym (t) + zm (t) ≤ Ma+b
∀ r ≥ 0.
0
Therefore, the sequences (ym )m≥0 and (zm )m≥0 converge to two functions y and z
that, for each t ∈ [0, ∞), satisfy
Z
Z t
1 r
A(s)g(s, y(s), z(s))ds dr,
y(t) = a +
0
0 A(r)
Z r
Z t
1
z(t) = b +
B(s)f (s, y(s), z(s))ds dr.
0 B(r)
0
Hence, y, z ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) and (y, z) is a solution of (1.4) satisfying
√ Z t q
−1
e
Λ(s)ds
,
a + g1 (b)g2 (a)SA (h2 )(t) ≤ y(t) ≤ Ma+b 2
0
Z
q
t
√
−1
e
b + f1 (a)f2 (b)SB (k2 )(t) ≤ z(t) ≤ Ma+b
2
Λ(s)ds
.
0
In the case where A(t) = B(t), we multiply the inequality
0
0
0
(A(t)(ym
(t) + zm
(t))) ≤ A(t)[g(t, ym (t), zm (t)) + f (t, ym (t), zm (t))]
0
0
by 2A(t)(ym
(t) + zm
(t)), we integrate on [0, t] and we proceed as below to obtain
Z ym (t)+zm (t)
2
0
0
e
(A(t)(ym
(t) + zm
(t))) ≤ (A(t))2 2Λ(t)
(φ(s) + ψ(s)) ds.
0
Which implies
0
0
ym
(t) + zm
(t) ≤
q
p
e
Λ(t)
2Φ(ym (t) + zm (t)).
So in this case (y, z) satisfies
a + ω1 (b)ω2 (a)SA (h2 )(t) ≤ y(t) ≤
−1
Ma+b
Z tq
0
e
Λ(s)ds
,
8
A. BEN DEKHIL
EJDE-2012/239
−1
b + ξ1 (a)ξ2 (b)SA (k2 )(t) ≤ z(t) ≤ Ma+b
Z t q
e
Λ(s)ds
, for t ≥ 0.
0
This completes the proof.
Proof of Theorem 1.5. We will show that (1.1) has a solution by finding a subsolution (u, v) and a supersolution (u, v), such that u ≤ u and v ≤ v. To do this, we
first prove the existence of a radial subsolution (u, v) to (1.1) by considering the
problem
4u = p1 (|x|)F (u + v),
in Rn
4v = q1 (|x|)G(u + v),
in Rn ,
with n ≥ 3. That is, (u, v) satisfies
1
(rn−1 u0 )0 = p1 (r)F (u + v), r ∈ (0, ∞)
rn−1
(2.1)
1
n−1 0 0
+
v)
r
∈
(0,
∞).
(r
v
)
=
q
(r)G(u
1
rn−1
By Theorem 1.3, we conclude that system (2.1) has a positive entire solution (u, v).
Now, since
4u = p1 (|x|)F (u + v) ≥ H(x, u, v),
in Rn ,
4v = q1 (|x|)G(u + v) ≥ K(x, u, v),
in Rn ,
we deduce that (u, v) is a subsolution of system (1.1).
Next we prove that (u, v) is bounded. Since (u, v) satisfies
(rn−1 u0 (r))0 =
rn−1 p1 (r)F (u(r) + v(r)),
(rn−1 v 0 (r))0 =
rn−1 q1 (r)G(u(r) + v(r)),
it follows that
(rn−1 (u0 (r) + v 0 (r)))0 = rn−1 [p1 (r)F (u(r) + v(r)) + q1 (r)G(u(r) + v(r))].
2(n−1)
(2.2)
Choose R > 0 so that r
(p1 (r) + q1 (r)) is nondecreasing on [R, ∞). Then,
after multiplying (2.2) by rn−1 (u0 (r)+v 0 (r)) and integrating from R to r, we obtain
2
rn−1 (u0 (r) + v 0 (r))
Z r
=C +2
t2(n−1) [p1 (t)F (u(t) + v(t)) + q1 (r)G(u(r) + v(r))]dt
R
≤ C + 2r2(n−1) p1 (r) + q1 (r)
Z r
[F 0 (u(t) + v(t)) + G 0 (u(t) + v(t))](u0 (t) + v 0 (t)) dt
×
R
≤ C + 2r2(n−1) (p1 (r) + q1 (r)) F(u(r) + v(r)) + G(u(r) + v(r)) ,
2
where C = Rn−1 (u0 (R) + v 0 (R)) . This yields
√
u0 (r) + v 0 (r)
1
2(F(u(r) + v(r)) + G(u(r) + v(r))) 2
√ 1−n
√ p
Cr
≤p
+ 2 p1 (r) + q1 (r).
(F(u(r) + v(r)) + G(u(r) + v(r)))
EJDE-2012/239
POSITIVE SOLUTIONS
9
Integrating the above inequality and using the fact that
F(u(r) + v(r)) + G(u(r) + v(r)) ≥ F(u(R) + v(R)) + G(u(R) + v(R)) = C1 ,
for all r ≥ R, and
p
p
p1 (r) + q1 (r) ≤ 2 r1+ε (p1 (r) + q1 (r))r−1−ε ≤ r1+ε (p1 (r) + q1 (r)) + r−1−ε ,
we obtain
Lα (u(r) + v(r))
r
√ Z r 1+ε
√
C
ε −1
(2 − n)(R2−n )−1 ,
≤ 2
s (p1 (s) + q1 (s))ds + 2 (εR ) +
C
1
R
where α = u(R) + v(R). Letting r → ∞, we deduce from hypothesis (H5) that
(u, v) is bounded. Thus, since (u, v) is nondecreasing, we have
lim u(r) = M1 > 0,
r→∞
lim v(r) = M2 > 0.
r→∞
Now, it is clear that (u, v) = (M1 , M2 ) is a supersolution for (1.1) and we have for
r ≥ 0,
u(r) ≥ M1 ≥ u(r), v(r) ≥ M2 ≥ v(r).
Hence the standard sub-supersolution method (see [7, 17]) implies that (1.1) has a
bounded solution (u, v) such that u ≤ u ≤ u and v ≤ v ≤ v. This completes the
proof.
Proof of Theorem 1.10. Let (u, v) be a positive entire large radial solution of (1.1).
Then (u, v) satisfies
(rn−1 u0 (r))0 ≤ rn−1 p1 (r)F (u(r) + v(r)),
(rn−1 v 0 (r))0 ≤ rn−1 q1 (r)G(u(r) + v(r)).
Adding these inequalities, we obtain
(rn−1 (u0 (r) + v 0 (r)))0 ≤ rn−1 p1 (r)F (u(r) + v(r)) + rn−1 q1 (r)G(u(r) + v(r))
≤ rn−1 (p1 (r) + q1 (r)) (F (u(r) + v(r)) + G(u(r) + v(r))) .
Multiplying the above inequality by 2rn−1 (u0 (r) + v 0 (r)) and integrating from 0 to
r, and using (H4), we obtain
Z r
2
rn−1 (u0 (r) + v 0 (r)) ≤ 2
s2(n−1) (p1 (s) + q1 (s)) F (u(s) + v(s))
0
+ G(u(s) + v(s)) (u0 (s) + v 0 (s)) ds
Z r
2(n−1)
≤ 2r
(p1 (r) + q1 (r))
F (u(s) + v(s))
0
+ G(u(s) + v(s)) (u0 (s) + v 0 (s)) ds
≤ 2r2(n−1) (p1 (r) + q1 (r)) (F(u(r) + v(r)) + G(u(r) + v(r)))
which implies
p
u0 (r) + v 0 (r)
p
≤ p1 (r) + q1 (r).
2(F(u(r) + v(r)) + G(u(r) + v(r)))
10
A. BEN DEKHIL
EJDE-2012/239
By integrating over [0, r], we obtain
Z
Lu(0)+v(0) (u(r) + (v(r)) ≤
r
p
p1 (s) + q1 (s)ds.
0
Letting r → ∞, we obtain that F and G satisfies the Keller-Osserman condition.
References
[1] N. Akhmediev, A. Ankiewicz; Partially coherent solitons on a finite background, Phys. Rev.
Lett. 82 (1999) 26612664.
[2] G. A. Afrouzi, T. A. Roushan; Existence of positive radial solutions for some nonlinear
elliptic systems, Bulletin of Mathematical analysis and Applications. 4 (2011) 146-154.
[3] F. Cirstea, V. Rădulescu; Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl. 81 (2002), 827-846.
[4] A. Ben Dekhil, N. Zeddini; Bounded and large radially symmetric solutions for some (p,q)Laplacian stationary systems, Electronic Journal of Differential Equations. Vol 2012, (2012)
no. 71 , 1-9.
[5] A. Ghanmi, H. Mâagli, V. Rădulescu, N. Zeddini; Large and bounded solutions for a class of
nonlinear Schrodinger stationary systems, Analysis and Applications. 4 (2009), 391-404.
[6] M. Ghergu, V. Rădulescu; Explosive solutions of semilinear elliptic systems with gradient
term, RACSAM Revista Real Academia de Ciencias (Serie A, Matemáticas) 97 (2003), 437445.
[7] J. Hernandez; Qualitative methods for nonlinear diffusion equations, Lecture Notes in Math;
1224, Springer, Berlin, 47-118, 1986.
[8] J. B. Keller; On solutions of ∆u = f (u), Communications on Pure and Applied Mathematics
10 (1957), 503-507.
[9] A. V. Lair; A necessary and sufficient condition for existence of large solutions to semilinear
elliptic equations, J.Math. Anal. Appl. 240 (1999), 205-218.
[10] A. V. Lair, A. W. Wood; Large solutions of sublinear elliptic equations, Nonlinear Anal.39
(2000) 745-753.
[11] A. V. Lair, A. W. Wood; Existence of entire large positive solutions of semilinear elliptic
systems, J. Diff. Equations. 164 (2000) 380-394.
[12] H. Li, P. Zhang, Z. Zhang: A remark on the existence of entir positive solutions for a class
of semilinear elliptic systems J. Math. Anal. Appl. 365 (2010) 338-341.
[13] C. R. Menyuk, Pulse propagation in an elliptically birefringent Kerr medium, IEEE J. Quantum Electron. 25 (1989) 2674-2682.
[14] R. Osserman, On the equality ∆u ≥ f (u), Pacific Journal of Mathematics 7 (1957), 1641-1647.
[15] Y. Peng, Y. Song; Existence of entire large positive solutions of a semilinear elliptic system,
Appl. Math. Comput. 155 (2004), 687-698.
[16] Yuan-Wei Qi; The existence and non-existence of positive solutions of elliptic systems in Rn ,
Methods and Applications of Analysis. 4 (2001) 557-568.
[17] D. Sattinger; Monotone methods in nonlinear elliptic and parabolic boundary value problems,
Indiana Univ. Math. 21 (1972), 979-1000.
[18] X. Wang and A. W Wood; Existence and nonexistence of entire positive solutions of semilinear elliptic systems, J. Math. Anal. Appl. 267 (2002),
[19] X. Zhang, L, Liu; The existence and nonexistence of entire positive solutions of semilineare
elliptic systems with gradient term, Journal of Mathematical Analysis and Applications. 371
(2010) 300-308.
[20] Zhijun Zhang; Existence of entire positive solutions for a class of semilinear elliptic systems,
Electronic Journal of Differential Equation. vol 2010 (2010), no. 16, 1-5.
[21] Zhijun Zhang, Yongxiu Shi, Yanxing Xue: Existence of entire solutions for semilinear elliptic
systems under the Keller-Osserman condition, Electronic Journal of Differential Equations.
vol. 2011 (2011) no. 39, 1-9.
Adel Ben Dekhil
Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El
Manar, Campus Universitaire, 2092 Tunis, Tunisia
E-mail address: Adel.Bendekhil@ipein.rnu.tn