Electronic Journal of Differential Equations, Vol. 2004(2004), No. 138, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
EXISTENCE RESULTS FOR ELLIPTIC SYSTEMS INVOLVING
CRITICAL SOBOLEV EXPONENTS
MOHAMMED BOUCHEKIF, YASMINA NASRI
Abstract. In this paper, we study the existence and nonexistence of positive
solutions of an elliptic system involving critical Sobolev exponent perturbed
by a weakly coupled term.
1. Introduction
We establish conditions for existence and nonexistence of nontrivial solutions to
the system
0
−∆u = (α + 1)uα v β+1 + µ(α0 + 1)uα v β
0
α+1 β
−∆v = (β + 1)u
v + µ(β + 1)u
u > 0, v > 0 in Ω
0
+1
α0 +1 β 0
v
in Ω
in Ω
(1.1)
u = v = 0 on ∂Ω,
where Ω is a bounded regular domain of RN (N ≥ 3) with smooth boundary ∂Ω,
µ ∈ R, α, β, α0 , β 0 are positive constants such that α + β = N 4−2 and 0 ≤ α0 + β 0 <
4
N −2 .
In the scalar case, the problem
−∆u = up + µuq in Ω
u > 0 in Ω
(1.2)
u = 0 on ∂Ω,
has been considered by several authors. The paper of Brezis-Nirenberg [7] has
drawn our attention.
In [7], they have obtained the following results: Suppose that Ω is a bounded
N +2
domain in RN , N ≥ 3, p = N
−2 , q = 1 and let λ1 > 0 denote the first eigenvalue
of the operator −∆ with homogeneous Dirichlet boundary conditions.
(1) If N ≥ 4, then for any µ ∈ (0, λ1 ) there exists a solution of (1.2).
(2) If N = 3, there exists µ∗ ∈ (0, λ1 ) such that for any µ ∈ (µ∗ , λ1 ) problem
(1.2) admits a solution.
2000 Mathematics Subject Classification. 35J20, 35J50, 35J60.
Key words and phrases. Elliptic system; critical Sobolev exponent; variational method;
moutain pass theorem.
c
2004
Texas State University - San Marcos.
Submitted July 6, 2004. Published November 25, 2004.
1
2
MOHAMMED BOUCHEKIF, YASMINA NASRI
EJDE-2004/138
(3) If N = 3 and Ω is a ball, then µ∗ = λ41 and for µ ≤ λ41 problem (1.2) has
no solution.
N +2
They have also obtained the following results for 1 < q < N
−2 :
(a) There is no solutions of (1.2) when µ ≤ 0 and Ω is a starshaped domain.
(b) When N ≥ 4, (1.2) has at least one solution for every µ > 0.
(c) When N = 3, We distinguish two cases:
(i) If 3 < q < 5, then for every µ > 0 there is a solution of (1.2).
(ii) If 1 < q ≤ 3, then for every µ large enough there is a solution of (1.2).
Moreover, (1.2) has no solution for every small µ > 0 when Ω is strictly
starshaped.
In the vectorial case, Alves et al. [1] and Bouchekif and Nasri [4] have extended
the results of [7] to elliptic system. A number of works contributed to study the
elliptic system for example: Boccardo and de Figueiredo [3], de Thélin and Vélin
[11] and Conti et al. [8].
Our aim is to generalize the results of [7] to an elliptic system when the lower
order perturbation of uα+1 v β+1 for each equation is weakly coupled i. e.
→
−∆U = ∇H + µ∇G,
where
→
∆=
0
∆
,
∆
H(u, v) = uα+1 v β+1 ,
U=
u
,
v
0
G(u, v) = uα +1 v β +1 and µ is a real parameter.
Our main results are stated as follows :
Theorem 1.1. If α + β = N 4−2 ; 0 ≤ α0 + β 0 <
domain, then (1.1) has no solution.
4
N −2 ;
Theorem 1.2. We suppose that N ≥ 4 and α + β =
0
0
µ ≤ 0 and Ω is a starshaped
4
N −2 .
We have:
4
N −2 ,
then for every µ > 0 problem (1.1) has at least one
• If 0 < α + β <
solution.
• If α0 + β 0 = 0, then for every 0 < µ < λ1 problem (1.1) has a solution.
Theorem 1.3. Assume that N = 3 and α + β = 4. We distinguish two cases:
• If 2 < α0 + β 0 < 4, then for every µ > 0 problem (1.1) has a solution.
• If 0 < α0 + β 0 ≤ 2, then for every µ large enough there exists a solution to
problem (1.1).
The paper is organized as follows. Section 2 contains some preliminaries and
notations. Section 3 contains the proof of nonexistence result. Section 4 deals with
the existence theorems proofs.
2. Preliminaries
Lemma 2.1 (Pohozaev identity [10]). Suppose that (u, v) ∈ [C 2 (Ω)]2 is the solution
to the problem
∂F
−∆u =
(u, v) in Ω
∂u
∂F
−∆v =
(u, v) in Ω
∂v
u = v = 0 on ∂Ω,
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ELLIPTIC SYSTEMS INVOLVING CRITICAL SOBOLEV EXPONENTS
3
where Ω is a bounded domain in RN (N ≥ 3) with smooth boundary ∂Ω, F ∈
C 1 (R2 ), F (0, 0) = 0, then we have
Z
Z
Z
∂v 2
∂F
∂u 2
∂F
(| | + | | )xνdσ + (N − 2)
(u
+v
)dx = 2N
F (u, v)dx (2.1)
∂ν
∂u
∂v
∂Ω ∂ν
Ω
Ω
where ν denotes the exterior unit normal.
We shall use the following version of the Brezis-Lieb lemma [6].
∂F
Lemma 2.2. Assume that F ∈ C 1 (RN ) with F (0) = 0 and | ∂u
| ≤ C|u|p−1 . Let
i
(un ) ⊂ Lp (Ω) with 1 ≤ p < ∞. If (un ) is bounded in Lp (Ω) and un → u a.e. on Ω,
then
Z
Z
F (un ) − F (un − u)) =
lim (
n→∞
Ω
F (u).
Ω
Let us define:
|∇u|2 dx
Ω
R
2
u∈H0 (Ω)\{0} (
|u|α+β+2 dx) α+β+2
Ω
R
(|∇u|2 + |∇v|2 )dx
Ω
= Sα,β (Ω) :=
inf 2
.
R
2
(u,v)∈[H01 (Ω)] \{(0,0)} (
|u|α+1 |v|β+1 dx) α+β+2
Ω
R
Sα+β+2 = Sα+β+2 (Ω) :=
Sα,β
inf
1
Lemma 2.3 ([1]). Let Ω be a domain in RN (not necessarily bounded) and α + β ≤
4
N −2 , then we have
h α + 1 β+1
−α−1 i
α + 1 α+β+2
α+β+2
Sα,β =
+
Sα+β+2 .
β+1
β+1
Moreover, if Sα+β+2 is attained at ω0 , then Sα,β is attained at (Aω0 , Bω0 ) for any
A
1/2
real constants A and B such that B
= ( α+1
.
β+1 )
We adopt the following notation:
R
1
• For p > 1, kukp = [ Ω |u|p dx] p ;
R
• H01 (Ω) is the Sobolev space endowed with the norm kuk1,2 = [ Ω |∇u|2 dx]1/2 ;
• k(u, v)k2E := kuk21,2 + kvk21,2 ;
• E := [H01 (Ω)]2 ;
• E 0 denotes the dual of E;
• 2∗ := N2N
−2 is the critical Sobolev exponent;
• u+ := max(u, 0) and u− = u+ − u.
The functional associated to problem (1.1) is written as
Z
Z
0
0
1
J(u, v) := k(u, v)k2E − (u+ )α+1 (v + )β+1 dx − µ (u+ )α +1 (v + )β +1 dx. (2.2)
2
Ω
Ω
3. Nonexistence result
Theorem 1.1 is a direct consequence of the Pohozaev identity.
Proof of Theorem 1.1. Arguing by contradiction. Suppose that problem (1.1) has
a solution (u, v) 6= (0, 0), applying Lemma 2.1 and putting
F (u, v) = H(u, v) + µG(u, v),
4
MOHAMMED BOUCHEKIF, YASMINA NASRI
EJDE-2004/138
the expression (2.1) becomes
Z
Z
0
0
∂u 2
∂v 2 0
0
| | + | | xν dσ = µ [2N − (N − 2)(α + β + 2)]
|u|α +1 |v|β +1 dx.
∂ν
∂ν
∂Ω
Ω
Since 2N − (N − 2)(α0 + β 0 + 2) > 0 and the fact that Ω is starshaped with respect
to the origin, we get
Z
∂u
∂v
0≤
(| |2 + | |2 )xν dσ < 0.
∂ν
∂ν
∂Ω
A contradiction. Hence (1.1) has no a solution for µ ≤ 0.
4. Existence results
The proof of Theorems 1.2 and 1.3 are based on the following AmbrosettiRabinowitz result [2].
Lemma 4.1 (Mountain Pass Theorem). Let J be a C 1 functional on a Banach
space E. Suppose there exits a neighborhood V of 0 in E and a positive constant ρ
such that
(i) J(u, v) ≥ ρ for every U in the boundary of V .
(ii) J(0, 0) < ρ and J(ϕ, ψ) < 0 for some Ψ := (ϕ, ψ) ∈
/ V.
We set
c = inf max J(φ(t))
φ∈Γ t∈[0,1]
with Γ = {φ ∈ C([0, 1], E) : φ(0) = 0, φ(1) = Ψ}. Then there exists a sequence
(un , vn ) in E such that J(un , vn ) → c and J 0 (un , vn ) → 0 in E 0 .
Proof. Using Holder’s inequality and Sobolev injection, we obtain that
Z
Z
0
0
1
2
+ α+1 + β+1
J(u, v) = k(u, v)kE − (u )
(v )
dx − µ (u+ )α +1 (v + )β +1 dx
2
Ω
Ω
1
α0 +β 0 +2
2
2∗
≥ k(u, v)kE − Ak(u, v)kE − Bk(u, v)kE
2
where A and B are positive constants.
If α0 + β 0 > 0 then (i) is satisfying for small norm k(u, v)kE = R. If α0 + β 0 = 0,
we have
∗
1
µ
J(u, v) ≥ 1 −
k(u, v)k2E − Ak(u, v)k2E
2
λ1
1−
µ
1
and condition (i) is still satisfied for µ < λ1 and R < ( 2Aλ1 ) 2∗ −2 . For any (ϕ, ψ) ∈
E with ϕ 6= 0 and ψ 6= 0, we have that limt→+∞ J(tϕ, tψ) = −∞. Thus, there
are many (ϕ, ψ) satisfying (ii). It will be important to use with a special (ϕ, ψ) :=
(t0 ϕ0 , t0 ψ0 ) for some t0 > 0 chosen large enough so that (ϕ, ψ) ∈
/ V , J(ϕ, ψ) < 0
2∗ Sα,β N/2
. Then there exists a sequence (un , vn ) ∈ E
and supt≥0 J(tϕ, tψ) < N ( 2∗ )
such that J(un , vn ) → c and J 0 (un , vn ) → 0 in E 0 .
Lemma 4.2. Suppose µ > 0 and let (un , vn ) be a sequence in E such that
J(un , vn ) → c and J 0 (un , vn ) → 0 in E 0 with
2∗ Sα,β N/2
2
Sα,β
( ∗ )
=
( ∗ )N/2
N 2
N −2 2
Then (un , vn ) is relatively compact in E.
c<
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ELLIPTIC SYSTEMS INVOLVING CRITICAL SOBOLEV EXPONENTS
5
Proof. We show that the sequence (un , vn ) is bounded in E. Since (un , vn ) satisfies:
1
k(un , vn )k2E −
2
Z
α+1 + β+1
(u+
(vn )
dx − µ
n)
Ω
Z
0
α +1 + β
(u+
(vn )
n)
0
+1
dx = c + o(1) (4.1)
Ω
and
k(un , vn )k2E − 2∗
Z
α+1 + β+1
(u+
(vn )
dx − µ(α0 + β 0 + 2)
n)
Z
Ω
0
α +1 + β
(u+
(vn )
n)
0
+1
dx
Ω
= hεn , (un , vn )i
(4.2)
with εn → 0 in E 0 . Combining (4.1) and (4.2), we obtain
Z
Z
α0 + β 0
2∗
α+1 + β+1
α0 +1 + β 0 +1
)
(v
)
dx
+
µ(
)
(u+
(vn )
dx
( − 1) (u+
n
n
n)
2
2
Ω
Ω
≤ c + o(1) + kεn kE 0 k(un , vn )kE .
(4.3)
From this inequality, we obtain
Z
α+1 + β+1
(u+
(vn )
dx ≤ C ,
n)
Ω
Z
α0 +1 + β 0 +1
(u+
(vn )
dx ≤ C .
n)
Ω
Where C is any generic positive constant. Therefore, the sequence (un , vn ) is
bounded in E. By the Sobolev embedding Theorem, there exists a subsequence
again denoted by (un , vn ) such that
• (un , vn ) → (u, v) weakly in E
• (un , vn ) → (u, v) strongly in Lr × Lq for 2 ≤ r, q < 2∗
• (un , vn ) → (u, v) a. e. on Ω.
2∗
β+1
Since wn := uα
and tn := uα+1
vnβ are bounded sequences in [L 2∗ −1 (Ω)]2 ,
n vn
n
α β+1
these sequences converge to w := u v
and to t := uα+1 v β respectively. Passing
to the limit, we obtain
0
−∆u = (α + 1)(u+ )α (v + )β+1 + µ(α0 + 1)(u+ )α (v + )β
0
0
+1
−∆v = (β + 1)(u+ )α+1 (v + )β + µ(β 0 + 1)(u+ )α +1 (v + )β
0
i.e
k(u, v)k2E
∗
Z
+ α+1
=2
(u )
+ β+1
(v )
0
Z
0
dx + µ(α + β + 2)
Ω
0
(u+ )α +1 (v + )β
0
+1
dx
Ω
Moreover,
J(u, v) = (
2∗
− 1)
2
Z
(u+ )α+1 (v + )β+1 dx + µ(
Ω
α0 + β 0
)
2
Z
0
(u+ )α +1 (v + )β
0
+1
dx ≥ 0.
Ω
We put
u = un + ϕn , v = vn + ψn
and H(un , vn ) = uα+1
vnβ+1
n
Applying Lemma 2.2 for H(un , vn ) and the following two relations (Brezis-Lieb [6])
kun k2 = ku − ϕn k2 = kuk2 + kϕn k2 + o(1) ,
kvn k2 = kv − ϕn k2 = kvk2 + kψn k2 + o(1),
6
MOHAMMED BOUCHEKIF, YASMINA NASRI
EJDE-2004/138
we obtain
1
J(u, v) + k(ϕn , ψn )k2E −
2
Z
+
H(ϕ+
n , ψn )dx = c + o(1)
(4.4)
Ω
and
k(ϕn , ψn )k2E
+
k(u, v)k2E
∗
Z
+
(H(u+ , v + ) + H(ϕ+
n , ψn ))dx
Ω
Z
(4.5)
0
0
+ α0 +1 + β 0 +1
+ µ(α + β + 2) (u )
(v )
dx + o(1).
=2
Ω
From this equality, we deduce
k(ϕn , ψn )k2E = 2∗
Z
+
H(ϕ+
n , ψn )dx + o(1).
Ω
We may therefore assume that
k(ϕn , ψn )k2E
→k
and 2
∗
Z
+
H(ϕ+
n , ψn )dx → k.
Ω
By the Sobolev inequality,
k(ϕn , ψn )k2E
≥ Sα,β
Z
ϕ+
n
α+1
22∗
(ψn+ )β+1 dx
.
Ω
In the limit, k ≥
We show that
E. Suppose that
∗
Sα,β ( 2k∗ )2/2 . It follows that
(un , vn ) → (u, v) strongly in
S
N/2
k ≥ 2∗ ( 2α,β
. Since
∗ )
J(u, v) +
S
N/2
.
either k = 0 or k ≥ 2∗ ( 2α,β
∗ )
E i. e. (ϕn , ψn ) → (0, 0) strongly in
k
=c
N
∗
S
(Ω)
k
≤ c i.e. c ≥ 2N ( α,β
)N/2 in contradiction with the
and J(u, v) ≥ 0, then N
2∗
hypothesis. Thus k = 0 and (un , vn ) → (u, v) strongly in E.
Proof of Theorem 1.2. It suffices to apply the mountain pass theorem with the
∗ S
(Ω) N/2
)
. We have to show that this geometric condition on c is
value c < 2N ( α,β
2∗
satisfied. Following the method in [7]. Without loss of generality we assume that
0 ∈ Ω, we use the test function
ϕ(x)
ωε (x) =
, ε>0
2 N −2
(ε + |x| ) 2
where ϕ is a cut-off positive function such that ϕ ≡ 1 in a neighborhood of 0. Let
A and B be positive constants such that
A
α + 1 1/2
=(
)
B
β+1
then (Aωε , Bωε ) is a solution of
−∆u = (α + 1)uα v β+1
in RN
−∆v = (β + 1)uα+1 v β
in RN
u(x) = 0,
v(x) = 0
as |x| → +∞
By [7, lemma 1], we obtain
sup J(tAωε , tBωε ) ≤
t≥0
N −2 2∗ Sα,β N/2
+ O ε 2 − µKεθ
∗
N 2
EJDE-2004/138
ELLIPTIC SYSTEMS INVOLVING CRITICAL SOBOLEV EXPONENTS
7
where K is a positive constant independent of ε, and θ := (4 − (α0 + β 0 )(N − 2))/4.
For θ < N 2−2 if N > 4 the inequality is satisfying for all 0 ≤ α0 + β 0 < N 4−2 .
Thus we obtain
2∗ Sα,β N/2
sup J(tAωε , tBωε ) <
for ε > 0 small enough.
N 2∗
t≥0
Then problem (1.1) has a solution for every µ > 0.
For N = 4, we distinguish two cases. Case 1: We have θ < 1 for all α0 + β 0 > 0.
Case 2: If α0 + β 0 = 0, we obtain
sup J(tAωε , tBωε ) ≤ (
t≥0
Sα,β 2
) + O(ε) − µKε| log ε|,
4
S
2
so for ε > 0 small enough, supt≥0 J(tAωε , tBωε ) < ( α,β
4 ) .
Note that the maximum principle ensures the positivity of solution.
Proof of Theorem 1.3. In three dimension the situation is different. We have
sup J(tAωε , tBωε ) ≤ 2(
t≥0
Sα,β 3/2
) + O(ε1/2 ) − µKεθ .
6
In this case we distinguish two cases.
(i) 0 < θ < 12 if 2 < α0 + β 0 < 4,
(ii) θ ≥ 12 if 0 < α0 + β 0 ≤ 2.
In case (i) we have the same conclusion as in the previous proof for (N ≥ 4).
So for the case 0 < α0 + β 0 ≤ 2, the existence of positive solution is assured for µ
S
3/2
. Thus (1.1) has a
large enough. It follows that supt≥0 J(tAωε , tBωε ) < 2( α,β
6 )
solution.
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8
MOHAMMED BOUCHEKIF, YASMINA NASRI
EJDE-2004/138
Mohammed Bouchekif
Departement of Mathematics, University of Tlemcen B. P. 119 Tlemcen 13000, Algeria
E-mail address: m bouchekif@mail.univ-tlemcen.dz
Yasmina Nasri
Departement of Mathematics, University of Tlemcen B. P. 119 Tlemcen 13000, Algeria
E-mail address: y nasri@mail.univ-tlemcen.dz