Electronic Journal of Differential Equations, Vol. 2002(2002), No. 83, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
On certain nonlinear elliptic systems with
indefinite terms ∗
Ahmed Bensedik & Mohammed Bouchekif
Abstract
We consider an elliptic quasi linear system with indefinite term on
a bounded domain. Under suitable conditions, existence and positivity
results for solutions are given.
1
Introduction
The purpose of this article is to find positive solutions to the system
∂H
(u, v) in Ω
∂u
∂H
(u, v) in Ω
−∆q v = m(x)
∂v
u = v = 0 on ∂Ω
−∆p u = m(x)
(1.1)
where Ω is a bounded regular domain of RN , with a smooth boundary ∂Ω,
∆p u := div(|∇u|p−2 ∇u) is the p-Laplacian with 1 < p < N , m is a continuous
function on Ω which changes sign, and H is a potential function which will be
specified later.
The case where the sign of m does not change has been studied by F. de
Thélin and J. Vélin [9]. These authors treat the system (1.1) with a function H
having the following properties
• There exists C > 0, for all x ∈ Ω, for all (u, v) ∈ D3 such that 0 ≤
0
0
H(x, u, v) ≤ C(|u|p + |v|q )
• There exists C 0 > 0, for all x ∈ Ω, for all (u, v) ∈ D2 such that H(x, u, v) ≤
C0
• There exists a positive function a in L∞ (Ω), such that for each x ∈ Ω and
(u, v) ∈ D1 ∩ R2+ , H(x, u, v) = a(x)uα+1 v β+1 ,
∗ Mathematics Subject Classifications: 35J20, 35J25, 35J60, 35J65, 35J70.
Key words: Elliptic systems, p-Laplacian, variational methods, mountain-pass Lemma,
Palais-Smale condition, potential function, Moser iterative method.
c
2002
Southwest Texas State University.
Work supported by research project B1301/02/2000.
Submitted April 2, 2002. Published October 2, 2002.
1
2
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
where
D1 = (u, v) ∈ R2 : |u| ≥ A or |v| ≥ A ,
D2 = (u, v) ∈ R2 \D1 : |u| ≥ δ or |v| ≥ δ ,
p
and D3 = R2 \(D1 ∪ D2 ) with A > δ > 0, 1 < p0 < p∗ := NN−p
, and 1 < q 0 < q ∗ .
They established the existence results under the conditions
α+1 β+1
+
>1
p
q
and
α+1 β+1
+
<1
p∗
q∗
by using a suitable application of the variational method due to AmbrosettiRabinowitz [2]. M. Bouchekif [4] generalized the work of F. de Thélin and
J.Velin [9] for the large class of functions of the form
H(u, v) = a|u|γ + c|v|δ + b|u|α+1 |v|β+1
where α, β ≥ 0; γ, δ > 1 and a, b and c are real numbers. The case where the
system (1.1) is governed by a single operator ∆p has been studied by Baghli [3].
Our aim is to extend to the system (1.1) the results obtained in the scalar
case (see [5]). Our existence results follow from modified quasilinear system in
order to apply the Palais-Smale condition (P.S.) and then the Moser’s Iterative
Scheme as in T. Ôtani [6] or in F. de Thélin and J. Vélin [9]. We consider only
weak solutions, and assume that H satisfies the following hypothesis.
(H1) H ∈ C 1 (R+ × R+ )
(H2) H(u, v) = o(up + v q ) as (u, v) → (0+ , 0+ )
(H3) There exists R0 > 0 and µ, 1 < µ < min(p∗ /p, q ∗ /q), such that
u ∂H
v ∂H
(u, v)+
(u, v) ≥ µH(u, v) > 0 ∀(u, v) ∈ R∗+ ×R∗+ , up +v q ≥ R0 .
p ∂u
q ∂v
2
Preliminaries and existence results
The values of H(u, v) are irrelevant for u ≤ 0 or v ≤ 0. We set
Z
Z
Z
1
1
I(u, v) =
|∇u|p dx +
|∇v|q dx −
m(x)H(u, v)dx
p Ω
q Ω
Ω
defined on E := W01,p (Ω) × W01,q (Ω). The solutions of the system (1.1) are
critical points of the functional I. Note that the functional I does not satisfy
in general the Palais-Smale condition since
Bµ H(u, v) :=
u ∂H
v ∂H
(u, v) +
(u, v) − µH(u, v)
p ∂u
q ∂v
EJDE–2002/83
Ahmed Bensedik & Mohammed Bouchekif
3
is not always bounded. In order to apply Ambrosetti-Rabinowitz Theorem [2],
we modify H so that the corresponding Bµ H(u, v) becomes bounded. Let
n H(u, v)
o
p
q
A(R) = max
:
R
≤
u
+
v
≤
R
+
1
(up + v q )µ
and
∂H
1
0
(u, v) + 2pµA(R)(R + 1)µ+1− p
sup |ηR
(r)|;
up +v q ≤R+1 ∂u
R≤r≤R+1
o
∂H
1
0
sup
(u, v) + 2qµA(R)(R + 1)µ+1− q
sup |ηR
(r)|
up +v q ≤R+1 ∂v
R≤r≤R+1
n
CR = max
sup
where ηR ∈ C 1 (R) is a cutting function defined by


= 1 if r ≤ R
ηR (r) < 0 if R < r < R + 1


= 0 if r ≥ R + 1.
Our main result is the following:
p∗ q ∗
∗ −p)(q ∗ −q) µ
(p
o(R
)
Theorem 2.1 Assume that (Hi )i=1,2,3 hold and CR =
for
R sufficiently large. Then the system (1.1) has at least one nontrivial solution
(u, v) in E ∩ [L∞ (Ω)]2 with u and v positive.
Before proving this theorem, we truncate the potential function H.
The modified problem
Let R ≥ R0 be fixed, and set
HR (u, v) := ηR (up + v q )H(u, v) + (1 − ηR (up + v q ))A(R)(up + v q )µ ,
By construction HR is C 1 and nonnegative. Let
0 p
MR :=(R + 1) p max
ηR (u + v q )(H(u, v) − A(R)(up + v q )µ )
q
u +v ≤R+1
+
max
up +v q ≤R+1
Bµ H(u, v),
Lemma 2.2 HR satisfies (H1)-(H3) and the following estimates
0 ≤ Bµ HR (u, v) ≤ MR , ∀(u, v) ∈ R+ × R+ ,
(2.1)
∂HR
(u, v) ≤ CR + µpA(R)up−1 (up + v q )µ−1 ,
∂u
(2.2)
∂HR
(u, v) ≤ CR + µqA(R)v q−1 (up + v q )µ−1 , ∀(u, v) ∈ R2+ ,
∂v
mR
HR (u, v) ≥ µ0 (up + v q )µ ∀(u, v) ∈ R∗+ × R∗+ , such that up + v q ≥ R0 ,
R0
(2.3)
with mR0 := min{H(u, v); up + v q = R0 }.
4
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
Proof. (H1) and (H2) can be easly verified for HR . We verify for (H3) as
follows: For any ν > 1,we have
Bν HR (u, v)
0
= (up + v q )ηR
(up + v q )[H(u, v) − A(R)(up + v q )µ ] + ηR (up + v q )Bν H(u, v),
for R0 ≤ up + v q ≤ R;
Bν HR (u, v) = Bν H(u, v) ≥ Bµ H(u, v) ≥ 0
for 1 < ν ≤ µ
for R ≤ up + v q ≤ R + 1;
Bν HR (u, v) ≥ ηR (up +v q )Bν H(u, v) ≥ ηR (up +v q )Bµ H(u, v) ≥ 0 for 1 < ν ≤ µ;
finally for up + v q ≥ R + 1, Bν HR (u, v) = 0 for any ν > 1. Thus (H3) holds for
HR .
Conditions (2.1) and (2.2) result from straightforward computations. Using
(H3), we have
HR (u, v) ≥
mR0 p
(u + v q )µ
R0µ
, ∀(u, v) ∈ R∗+ × R∗+ such that up + v q ≥ R0 . (2.4)
1
In fact, put f (t) := HR (t1/p u, t q v) with up + v q ≥ R0 then
1
i
1
1
t q v ∂HR 1/p
1 h t1/p u ∂HR 1/p
(t u, t q v)
(t u, t q v) +
q ∂v
t
p
∂u
µ
R0
≥ f (t) for all t ≥ t0 := p
(≤ 1).
t
u + vq
f 0 (t) =
(2.5)
Integrating (2.5) between t0 and t, we obtain
f (t)
tµ
≥ µ
f (t0 )
t0
for all t ≥ t0
(2.6)
and taking t = 1 in (2.6), we have
HR (u, v) = f (1) ≥
(up + v q )µ
f (t0 )
R0µ
1/p
0
and f (t0 ) = HR (u1 , v1 ) = H(u1 , v1 ), where u1 = ( upR+v
u, and
q)
R0
1/q
v1 = ( up +vq ) v. Consequently,
min
up +v q ≥R0
f (t0 (u, v)) =
min
up +v q =R0
H(u, v),
hence (2.4) follows. Now, consider the modified system
∂HR
(u, v) in Ω
∂u
∂HR
−∆q v = m(x)
(u, v) in Ω
∂v
u = v = 0 on ∂Ω
−∆p u = m(x)
(2.7)
EJDE–2002/83
Ahmed Bensedik & Mohammed Bouchekif
5
which has an associated functional IR defined on E as
Z
Z
Z
1
1
IR (u, v) =
|∇u|p dx +
|∇v|q dx −
m(x)HR (u, v)dx.
p Ω
q Ω
Ω
Lemma 2.3 Under the hypotheses (H1)-(H3), the functional IR satisfies the
Palais-Smale condition.
Proof. Let (un , vn ) be an element of E such that IR (un , vn ) is bounded and
0
0
0
IR
(un , vn ) → 0 strongly in W0−1,p (Ω) × W0−1,q (Ω) (dual space of E).
Claim 1. (un , vn ) is bounded in E. In fact, for any M , we have
Z
Z
Z
1
1
−M ≤
|∇un |p dx +
|∇vn |q dx −
m(x)HR (un , vn )dx ≤ M ;
p Ω
q Ω
Ω
and for ε ∈ (0, 1), we have again
Z
Z
1
1
−ε ≤
|∇un |p dx +
|∇vn |q dx
p Ω
q Ω
Z
un ∂HR
vn ∂HR
(un , vn ) +
(un , vn ) dx ≤ ε.
−
m(x)
p ∂u
q ∂v
Ω
Then we obtain
Z
Z
Z
µ−1
µ−1
|∇un |p dx +
|∇vn |q dx ≤ M µ −
m(x)Bµ HR (u, v)dx
p
q
Ω
Ω
Ω
≤ M µ + 1 + |m|0 MR (meas Ω)
where |m|0 := max
−
x∈Ω
(|m(x)|). Hence (un , vn ) is bounded in E.
Claim 2. (un , vn ) converges strongly in E. Since (un , vn ) is bounded in E,
there exists a subsequence denoted again by (un , vn ) which converges weakly in
E and strongly in the space Lζ (Ω)×Lη (Ω) for any ζ and η such that, 1 < ζ < p∗
0
and 1 < η < q ∗ . ¿From the definition of IR
, we write
Z
(|∇un |p−2 ∇un − |∇ul |p−2 ∇ul )∇(un − ul )dx
Ω
0
0
= hIR
(un , vn ) − IR
(ul , vl ), (un − ul , 0)i
Z
i
h ∂H
∂HR
R
(un , vn ) −
(ul , vl ) (un − ul )dx.
+
m(x)
∂u
∂u
Ω
0
0
0
By assumptions on IR
, hIR
(un , vn ) − IR
(ul , vl ), (un − ul , 0)i converges to 0 as
n and l tend to +∞. In what follows, C denotes a generic positive constant.
Now, we prove that
Z
∂HR
∂HR
Cn,l :=
m(x)[
(un , vn ) −
(ul , vl )](un − ul )dx
∂u
∂u
Ω
6
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
converges to 0 as n and l tend to +∞. We have
Z
∂HR
∂HR
|Cn,l | ≤ |m|0 [|
(un , vn )| + |
(ul , vl )|]|un − ul |dx
∂u
∂u
Ω
and
Z
∂HR
(un , vn )||un − ul |dx
∂u
Ω
Z
≤
(CR + µpA(R)|un |p−1 (|un |p + |vn |q )µ−1 )|un − ul |dx
Ω
Z
µ−1
≤ 2
CR (1 + |un |µp−1 + |un |p−1 |vn |qµ−q )|un − ul |dx
Ω
Z
hZ
µ−1
≤ 2
CR
|un − ul |dx +
|un |µp−1 |un − ul |dx
Ω
Ω
Z
i
p−1
qµ−q
+
|un | |vn |
|un − ul |dx .
|
Ω
Using Hölder’s inequality and Sobolev’s embeddings, we obtain
Z ∂HR
(un , vn )|un − ul |dx
∂u
Ω
hZ
i1/p
p−1
≤ 2µ−1 CR (meas Ω) p
|un − ul |p dx
Ω
+2µ−1 CR
hZ
1
i µp−1
hZ
i µp
µp
|un |µp dx
|un − ul |µp dx
+2µ−1 CR
hZ
1
i p−1
hZ
i µ−1
hZ
i µp
µp
µ
|vn |µq dx
|un − ul |µp dx
,
|un |µp dx
Ω
Ω
Ω
Ω
Ω
(because (un ) ∈ W01,p (Ω) and µp < p∗ , (vn ) ∈ W01,q (Ω) and µq < q ∗ ). Then
Z ∂HR
(un , vn )|un − ul |dx ≤ Ckun − ul kLp (Ω) + Ckun − ul kLµp (Ω) .
∂u
Ω
Similarly, we obtain
Z
∂HR
|
(ul , vl )||un − ul |dx ≤ Ckun − ul kLp (Ω) + Ckun − ul kLµp (Ω) ,
∂u
Ω
and so |Cn,l | ≤ |m|0 (Ckun − ul kLp + Ckun − ul kLµp ). Hence Cn,l converges to
0 as n and l tend to +∞.
We have the following algebraic relation [8]
|∇un − ∇ul |p
s/2
1− 2s
≤ C (|∇un |p−2 ∇un − |∇ul |p−2 ∇ul )∇(un − ul )
|∇un |p + |∇ul |p
,
(2.8)
EJDE–2002/83
Ahmed Bensedik & Mohammed Bouchekif
7
(
p for 1 < p ≤ 2
. Integrating (2.8) on Ω, and using Hölder’s inwhere s =
2 for 2 < p
equality in the right hand side, we obtain
kun − ul kp1,p
hZ
i 2s
1− 2s
≤C
(|∇un |p−2 ∇un −|∇ul |p−2 ∇ul )∇(un −ul )dx
kun kp1,p +kul kp1,p
.
Ω
Now since
Z
(|∇un |p−2 ∇un − |∇ul |p−2 ∇ul )∇(un − ul )dx → 0
Ω
as n and l tend to +∞, the sequence (un ) converges strongly in W01,p (Ω). Similarly we prove that the sequence (vn ) converges strongly in W01,q (Ω).
The next lemma shows that IR satisfies the geometric assumptions of the
Mountain-Pass Theorem.
Proposition 2.4 Under assumptions (H1)-(H3) we have
1. There exist two positive real numbers ρ, σ and a neighborhood Vρ of the
origin of E, such that for any element (u, v) on the boundary of Vρ :
IR (u, v) ≥ σ > 0.
2. There exist (φ, θ) in E such that IR (φ, θ) < 0.
Proof. From (H2) and taking into account that HR (u, v) = H(u, v) for up +
v q ≤ R, we can write
∀ε > 0, ∃δε > 0 : up + v q ≤ δε =⇒ HR (u, v) ≤ ε(up + v q ),
and since HR (u, v)/(up + v q )µ is uniformly bounded as up + v q tends to +∞
∃M (ε, R) > 0 : up + v q ≥ δε =⇒ HR (u, v) ≤ M (up + v q )µ .
Then for every (u, v) in R+ × R+ we have
HR (u, v) ≤ ε(up + v q ) + M (up + v q )µ .
Hence
Z
m(x)HR (u, v)dx
Z
h Z
i
p
q
≤ |m|0 ε (u + v )dx + M (up + v q )µ dx
Ω
Ω
Z
hZ
i
p
µ−1
pµ
≤ |m|0
(εu + 2
M u )dx + (εv q + 2µ−1 M v qµ )dx
Ω
Ω
µq ≤ C|m|0 ε(kukp1,p + kvkq1,q ) + M (kukµp
1,p + kvk1,q ) .
Ω
8
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
For IR (u, v), we obtain
IR (u, v) ≥
1
− C|m|0 (ε + M kukµp−p
1,p )
p
1
+kvkq1,q − C|m|0 (ε + M kvkµq−q
1,q ) ≥ σ > 0,
q
kukp1,p
for every (u, v) in the sphere S(0, ρ) of E where ρ is such that 0 < ρ < min(ρ1 , ρ2 )
with
ρ1 =
1
1
ε µp−p
−
pM C|m|0
M
and ρ2 =
1
ε µq−q
1
−
qM C|m|0
M
with ε sufficiently small.
2. Choose (φ, θ) ∈ E such that: φ > 0, θ > 0,
supp φ ⊂ Ω+ ,
supp θ ⊂ Ω+ ,
where Ω+ = {x ∈ Ω; m(x) > 0}. Hence, for t sufficiently large,
Z
t
t
kφkp1,p + kθkq1,q −
m(x)HR (t1/p φ, t1/q θ)dx
IR (t1/p φ, t1/q θ) =
p
q
Ω
Z
kφkp1,p
kθkq1,q µ mR0
≤ t
+
−t
m(x)(φp + θq )µ dx
p
q
R0µ Ω
and so limt→+∞ IR (t1/p φ, t1/q θ) = −∞, (because µ > 1). By continuity of IR
on E, there exists (φ, θ) in E \ B(0, ρ) such that IR (φ, θ) < 0. By the usual
Mountain-Pass Theorem, we know that there exists a critical point of IR which
we denote by (uR , vR ), and corresponding to a critical value cR ≥ σ. Since
+
+
HR
(u+
R , vR ), where uR := max(uR , 0), is also solution for the system (Sp,q ), we
assume uR ≥ 0 and vR ≥ 0. Positivity of uR and vR follows from Harnack’s inequality (see J. Serrin [7]). We prove now that, under some conditions, (uR , vR )
is also solution of the system (2.7).
3
Existence results
We adapt the Moser iteration used in [6, 9] to construct two strictly unbounded
sequences (λk )k∈N and (µk )k∈N such that (uR , vR ) satisfies
uR ∈ Lλk (Ω)
uR ∈ Lλk+1 (Ω)
if
then
vR ∈ Lµk (Ω)
vR ∈ Lµk+1 (Ω).
Bootstrap argument
Proposition 3.1 Under the assumptions of Theorem 2.1, there exist two sequences (λk )k and (µk )k such that
1. For each k, uR and vR belong to Lλk (Ω) and Lµk (Ω) respectively
EJDE–2002/83
Ahmed Bensedik & Mohammed Bouchekif
9
2. There exist two positive constants Cp and Cq such that
kuR k∞ ≤ lim sup kuR kLλk ≤ Cp ,
and
k→+∞
kvR k∞ ≤ lim sup kvR kLµk ≤ Cq .
k→+∞
Lemma 3.2 Let (ak )k∈N and (bk )k∈N be two positive sequences satisfying, for
each integer k, the relations
q(µ − 1)
p + ak
+
= 1,
λk
µk
q + bk
p(µ − 1)
+
= 1.
µk
λk
and
(3.1)
If uR and vR are in Lλk (Ω) and Lµk (Ω) respectively, λk+1 ≤ (1+ apk )πp , µk+1 ≤
(1 + bqk )πq with 1 < πp < p∗ and 1 < πq < q ∗ , then we have:
λ
n a
o k+1
ak λ
λk
µk 1/p 1+ pk
kuR kλk+1
≤
K
θ
1
+
C
|m|
(ku
k
+
kv
k
)
,
p
p
R
0
R
R
µk
λk
k+1
p
µk+1
n 1 o b
bk λk
µk+1
µk q 1+ qk
CR |m|0 (kuR kλk + kvR kµk )
kvR kµk+1 ≤ Kq θq 1 +
q
(3.2)
(3.3)
where kzkβ is kzkLβ (Ω) and Kp , Kq , θp , and θq are positive constants.
Proof. Remark that if, for an infinite number of integers k, kuR kλk ≤ 1 then
kuR k∞ ≤ 1 and proposition 1 is proved. So we suppose that kuR kλk ≥ 1 for all
k ∈ N. Let ζn , n ∈ N, be C 1 functions such that
ζn (s) = s
if s ≤ n
ζn (s) = n + 1 if s ≥ n + 2
0 < ζn0 (s) < 1 if s ∈ R+ .
k
∈ W01,p (Ω) ∩ L∞ (Ω) and uR satisfies the first
Put un := ζn (uR ), then u1+a
n
k
equation of the system (2.7). Multiply this equation by u1+a
and integrate
n
over Ω to get
Z
Z
∂HR
k
k
−∆p uR .u1+a
dx
=
m(x)
(uR , vR )u1+a
dx
n
n
∂u
Ω
Ω
Z
qµ−q 1+ak
≤ 2µ−1 CR |m|0 (1 + upµ−1
+ up−1
)un dx.
R
R vR
Ω
Since un ≤ uR , we have
Z
k
−∆p uR .u1+a
dx
n
Ω
≤2
µ−1
CR |m|0
nZ
Ω
k
u1+a
dx
R
+
Z
Ω
k
upµ+a
dx
R
+
Z
Ω
o
k qµ−q
up+a
vR dx .
R
Using Hölder’s inequality, we obtain
Z
n
1+a
1− λ k
1+ak
µ−1
k
k
k ku k
−∆p uR .u1+a
dx
≤
2
C
|m|
+ kuR kpµ+a
R λk
R
0 (meas Ω)
n
pµ+ak
Ω
o
k
+kuR kp+a
kvR kqµ−q
.
µk
λk
10
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
We shall show below that pµ + ak = λk . Since kuR kλk ≥ 1, we get
Z
−∆p uR .un1+ak dx
Ω
k
≤ 2µ−1 CR |m|0 max(1, meas Ω) 2kuR kλλkk + kuR kλp+a
kvR kqµ−q
.
µk
k
Moreover, using the relation (3.1), we obtain
k
kuR kp+a
kvR kqµ−q
≤ kuR kλλkk + kvR kµµkk ,
µk
λk
so, with c0 := 3 max(1, meas Ω),
Z
k
−∆p uR .u1+a
dx ≤ 2µ−1 c0 CR |m|0 kuR kλλkk + kvR kµµkk .
n
(3.4)
Ω
On the other hand we have
Z
k
−∆p uR .u1+a
dx =
n
(1 + ak )
Ω
≥ (1 + ak )
Z
|∇uR |p ζn0 (uR )uank dx
ZΩ
|∇uR |p (ζn0 (uR ))p uank dx
Ω
=
Z
(1 + ak )
|∇un |p uank dx
Ω
and thus
Z
k
−∆p uR .u1+a
dx
n
Ω
1+
≥
Z
|∇un |p uank dx .
(3.5)
Ω
ak
Since un p ∈ W01,p (Ω), the continuous imbedding of W01,p (Ω) in Lπp (Ω) implies
the existence of a positive constant c such that
Z
π1
1/p
Z
a
a
1+ pk πp
1+ k
p
| un
|dx
≤ c
|∇un p |p dx
Ω
Ω
Z
1/p
ak uank |∇un |p dx
.
(3.6)
= c 1+
p
Ω
By assumption, we have λk+1 ≤ jk := 1 + apk πp . Then
kun kλk+1 ≤ (meas Ω)mk kun kjk ,
and thus
λ
where mk :=
1
1
−
λk+1
(1 + apk )πp
λ
kun kλk+1
≤ Kp kun kjkk+1
k+1
where Kp is a positive constant greater than (meas Ω)mk λk+1 independently of
the integer k. By the relation (3.6),
λ
kun kjkk+1
Z
1/p i λk+1
a
ak 1+ k
ak
p
p .
≤ c[1 + ]
un |∇un | dx
p
Ω
h
(3.7)
EJDE–2002/83
Ahmed Bensedik & Mohammed Bouchekif
11
Combining the inequalities (3.4)-(3.7), we deduce
λ
kun kλk+1
k+1
with θp = 2
show (3.3).
µ−1
p
λk+1
1/p 1+ apk
ak λk
µk
CR |m|0 (kuR kλk + kvR kµk )
,
≤ Kp θ p 1 +
p
1/p
c0 c. Hence, by letting n → +∞, we obtain (3.2). Similarly we
Construction and definition of (λk )k and (µk )k .
Here we construct the sequences (λk )k and (µk )k using tools similar as those
in [O] or [TV]. The first terms of each sequence cannot be determined directly
by using the Rellich-Kondrachov continuous imbedding result. So, we first conbk )k and (b
struct two other sequences (λ
µk )k , such that for each k, uR and vR
bk
λ
µ
bk
bk and
belong to L (Ω) and L (Ω) respectively. By a suitable choice of k0 , λ
0
µ
bk0 determine the first terms of (λk )k and (µk )k
bk )k and (b
Construction of (λ
µk )k .
Suppose p ≤ q, and fix a number s, such that cp/p∗ < s < 1/µ. Put
∗
b := 1 + s p .
C
2 2 p
bk = µpC
b > 1, 1 < µpC
b < p∗ and 1 < µq C
b < q ∗ . Now, we take λ
bk
Remark that C
k
b . By definition of (a ), we have
and µ
bk = µq C
k
p + ak
µ−1
+
q=1
bk
µ
bk
λ
bk − pµ. Similarly, we find bk = µ
then ak = λ
bk − qµ.
Lemma 3.3 For each integer k, uR ∈ Lλk (Ω) and vR ∈ Lµbk (Ω).
b
b0 = µp < p∗ , µ
Proof. By induction. For k = 0, λ
b0 = µq < q ∗ , and since
b
(uR , vR ) ∈ E, by the Sobolev imbedding theorem, we have uR ∈ Lλ0 (Ω) and
vR ∈ Lµb0 (Ω).
Suppose that the proposition is true for all integers k 0 such that 0 ≤ k 0 ≤ k.
Take
b and πq = µq C.
b
πp = µpC
Since uR ∈ Lλk (Ω) and
b
[1 +
bk − µp ak
λ
b = µpC
b + µ2 pC
b k+1 − µ2 pC
b ≥ µpC
b k+1
]πp = 1 +
µpC
p
p
bk+1 , Lemma 3 allows us to write uR ∈ Lλbk+1 (Ω) and vR ∈
i.e. [1 + apk ]πp ≥ λ
Lµbk+1 (Ω).
12
Nonlinear elliptic systems with indefinite terms
Construction of (λk )k and (µk )k
C=
N
,
N −p
EJDE–2002/83
Put
and δ =
p
b k0 − (µ − 1) C,
µC
N
where the integer k0 is chosen so as to have δ > 0. The sequences (λk )k and
(µk )k are defined by λk = pfk and µk = qfk , where
fk =
C
[δC k−1 + (µ − 1)].
C −1
We remark that the three last sequences are strictly increasing and unbounded.
Furthermore (fk ) satisfies the relation fk+1 = C[fk − (µ − 1)].
Proof of Proposition 2. 1. We show by induction that for all integer k,
uR ∈ Lλk (Ω) and vR ∈ Lµk (Ω). For k = 0,
λ0 = pf0 =
pC δ
N p b k0 b
+ (µ − 1) = p
µC
= λ k0 ,
C −1 C
p N
and similarly, µ0 = µ
bk0 .
By Lemma 4, uR ∈ Lλ0 (Ω) and vR ∈ Lµ0 (Ω). Suppose that (uR , vR ) ∈
Lλk (Ω) × Lµk (Ω). First we establish that λk = ak + pµ. By condition (3.1),
1=
thus
p + ak
µ−1
p
q
ak
q
+q
=
−
+
+µ ,
λk
µk
λk
µk
λk
µk
ak
µ
+
=1
pfk
fk
which implies ak = p(fk − µ) = λk − pµ, and similarly µk = bk + qµ = q(fk − µ).
Now when we take πp = Cp and πq = Cq, we then have
ak πp = (1 + fk − µ)Cp = pfk+1 = λk+1 .
1+
p
and similarly [1+ bqk ]πq = µk+1 . Since (uR , vR ) ∈ Lλk (Ω)×Lµk (Ω), we conclude,
according to Lemma 3, that
(uR , vR ) ∈ Lλk+1 (Ω) × Lµk+1 (Ω).
So uR ∈ Lλk (Ω), and vR ∈ Lµk (Ω), for all integer k.
2. Now we prove that uR and vR are bounded. By Lemma 3, we have
λ
n ak
1/p o 1+k+1
ak λ
λk
µk
p ,
kuR kλk+1
≤
K
θ
C
|m|
(ku
k
1
+
+
kv
k
)
R
0
R
R
p
p
µk
λk
k+1
p
µk+1
n q1 o 1+ bk
bk λk
µk
q .
kvR kµµk+1
≤
K
θ
1
+
C
|m|
(ku
k
+
kv
k
)
R
0
R λk
R µk
q
q
k+1
q
EJDE–2002/83
We remark that
Ahmed Bensedik & Mohammed Bouchekif
λk+1
= pC
1 + apk
and
13
µk+1
= qC.
1 + bqk
Consequently,
ak pC
λ
kuR kλk+1
≤ 2C Kp θppC 1 +
(|m|0 CR )C max kuR kλλkk C , kvR kµµkk C ,
∞
k+1
p
bk qC
µk+1
C
qC
kvR kµk+1 ≤ 2 Kq θq 1 +
(|m|0 CR )C max kuR kλλkk C , kvR kµµkk C .
q
We have
1+
ak
bk
C δ
=1+
= 1 + fk − µ <
+ µ − 1 Ck.
p
q
C −1 C
Take
C δ
+ µ − 1 [Kp + Kq ]
C −1 C
p q
and θ := 2|m|0 max(θp , θq ), then we can write
λ
Cλk
µk+1
q C kqC C
Cµk
max kuR kλk+1
,
kv
k
≤
(A
θ)
C
C
max
ku
k
,
kv
k
.
R
R
R
µ
R
µ
λk
k+1
k
k+1
A :=
We construct an iterative relation
Ek+1 ≤ rk + CEk
where Ek = ln max(kuR kλλkk , kvR kµµkk ), and rk = ak + b, with a = ln C qC and
b = ln[Aq θCR ]C . Proceeding step by step, we find
Ek+1
≤ rk + Crk−1 + C 2 rk−2 + · · · + C k r0 + C k+1 E0,
Ek+1
≤ C k+1 E0 +
k
X
C i rk−i .
i=0
Let us evaluate
σk :=
k
X
C i rk−i .
i=0
We have rk−i = a(k − i) + b = ak + b − ai, then
σk
k
X
i
C −a
k
X
iC i
=
(ak + b)
=
bC k+2 + (a − b)C k+1 + (1 − C)ak − [C(a + b) − b]
.
(C − 1)2
i=0
i=0
Since C > 1, and a, b are positive, we have
σk ≤
bC k+2 + (a − b)C k+1
(C − 1)2
14
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
then
a−b
bC k+2
+ C k+1
+ E0 .
2
2
(C − 1)
(C − 1)
By an appropriate choice for the constants Kp and Kq , we ensure that
Ek+1 ≤
b−a
≥ E0 .
(C − 1)2
Recall that
b − a = C ln
Aq θCR
Cq
with A =
C δ
+ µ − 1 [Kp + Kq ];
C −1 C
hence Ek+1 ≤ bC k+2 /(C −1)2 . By the definition of Ek+1 and the last inequality,
we obtain
bC k+2
,
λk+1 ln kuR kλk+1 ≤ Ek+1 ≤
(C − 1)2
thus
bC k+2
ln kuR kλk+1 ≤
.
λk+1 (C − 1)2
Letting k → +∞, we find
ln kuR k∞ ≤
bC
,
pδ(C − 1)
or
ln kuR k∞ ≤
N
b.
δp2
Similarly
N
b.
δq 2
We deduce the existence of constants Cp and Cq such that:
ln kvR k∞ ≤
kuR k∞ ≤ Cp
and kvR k∞ ≤ Cq .
Take
N
N
b, and Cq = exp 2 b.
δp2
δq
Then Cp and Cq , are greater than 1, which is compatible with the remark
noted at the beginning of the proof of Lemma 3. This completes the proof of
proposition 1.
Cp = exp
Proof of Theorem 2.1. If kuR kp∞ + kvR kq∞ < R, then (uR , vR ) furnishes a
solution of the system (1.1). We have
kuR kp∞ + kvR kq∞ ≤ Cpp + Cqq ≤ 2 exp
N
b;
δp
N
so it is sufficient to have 2 exp δp
b < R for R large enough, to get (uR , vR )
solution of the initial system (1.1). Replacing b by its expression, we obtain
CN
(Aq θCR ) δp < R2 i.e.
δp
R CN
CR < δp
.
2 CN θAq
EJDE–2002/83
Ahmed Bensedik & Mohammed Bouchekif
15
But δ can be chosen such that
δp
N2
p∗ q ∗
>
µ= ∗
µ
CN
pq
(p − p)(q ∗ − q)
p∗ q ∗
and we can take CR <
if
R (p∗ −p)(q∗ −q)
δp
2 CN
θAq
µ
. Then (uR , vR ) is solution of system (1.1)
p∗ q ∗
CR = o R (p∗ −p)(q∗ −q) µ
for R sufficiently large.
Examples
Now, we present functions
satisfying the hypotheses in our main result.
∗
∗
For 1 < γ < min( pp , qq ), let
H(u, v) = (up + v q )γ
be defined on R2+ . Then H satisfies the hypotheses of Theorem 2.1.
β+1
β+1
α+1
For α, β ≥ 0, α+1
p + q > 1 and p∗ + q ∗ < 1, let
H(u, v) = uα+1 v β+1 .
be defined on R2+ . Then H satisfies the hypotheses of Theorem 2.1.
References
[1] Adams, R. A., Sobolev spaces, Academic press, New York, 1975.
[2] Ambrosetti, A. P.H. Rabinowitz, Dual variational methods in critical point
theory and application, J. of Funct. Anal., 14 (1973), 349-381.
[3] Baghli, S., Résultats d’existence de solutions non négatives de classe de
systèmes elliptiques contenant le p-Laplacien, Thèse de Magister, no 445
(1998), Université de Tlemcen, Algérie.
[4] Bouchekif, M., Some Existence results for a class of quasilinear elliptic systems, Richerche di Matematica, vol. XLVI, fas. 1 (1997), 203-219.
[5] Bouchekif, M., On certain quasilinear elliptic equations with indefinite terms,
Funkcialaj Ekvacioj, 41 (1998), 309-316.
[6] Ôtani, T., Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 1083-1141.
[7] Serrin, J., Local behaviour of solutions of quasilinear equations, Acta. Math.,
113 (1964), 302-347.
16
Nonlinear elliptic systems with indefinite terms
EJDE–2002/83
[8] Simon, J., Régularité de la solution d’une equation non linéaire dans RN ,
Ph. Benilian and J. Robert eds. Lecture notes in math. 665, Springer Verlag,
Berlin, (1978), 205-227.
[9] De Thélin, F. and J.Vélin, Existence and nonexistence of nontrivial solutions
for some nonlinear elliptic systems, Revista Matematica, 6 (1993), 153-193.
Ahmed Bensedik (e-mail: ahmed benseddik2002@yahoo.fr)
Mohammed Bouchekif (e-mail: m bouchekif@mail.univ-tlemcen.dz)
Département de Mathématiques, Université de Tlemcen
B. P. 119, 13000 Tlemcen, Algérie.