Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 4(2011), Pages 1-14.
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR
ELLIPTIC SYSTEMS WITH INDEFINITE WEIGHTS
(COMMUNICATED BY VICENTIU RADULESCU)
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
Abstract. We establish the existence of a nontrivial solution of system:

p−2 + λ0 c(x)u|u|α−1 |v|β+1 + f
in Ω
 −∆p u = λ a(x)u|u|
−∆q v = µ b(x)v|v|q−2 + λ0 c(x)|u|α+1 v|v|β−1 + g
in Ω

(u, v) ∈ W01,p (Ω) × W01,q (Ω)
under some restrictions on λ, µ, λ0 , α, β, f and g. We show this result by a local
minimization.
1. Introduction
The purpose of this paper is to investigate the existence of a solution of the
system:

 −∆p u = λ a(x)u|u|p−2 + λ0 c(x)u|u|α−1 |v|β+1 + f in Ω
−∆q v = µ b(x)v|v|q−2 + λ0 c(x)|u|α+1 v|v|β−1 + g
in Ω
(1.1)

(u, v) ∈ W01,p (Ω) × W01,q (Ω)
where Ω ⊂ Rn is a bounded domain, n ≥ 3, 1 < p, q < n, α > −1, β >
−1, λ, µ and λ0 are positive parameters, functions a(x), b(x) and c(x) ∈ C(Ω) are
smooth functions with change sign on Ω. For all p ≥ 1 ∆p u is the p-Laplacian defined by ∆p u = div(|∇u|p−2 ∇u) and W01,p (Ω) is the closure of C0∞ (Ω) with respect
to the norm kuk1,p := k∇ukp , where k.kp represent the norm of Lebesgue space
Lp (Ω). Let p0 be the conjugate to p, W0−1,p0 (Ω) is the dual space to W01,p (Ω) and
we denote by k.k−1,p0 its norm. We denote by < x∗ , x >X ∗ ,X the natural duality
pairing between X and X ∗ .
∗
For all p > 1, Sp = inf { k∇ukpp ; kukpp∗ = 1 u ∈ W01,p (Ω)} is the best Sobolev
∗
np
constant of immersion W 1,p (Ω) ,→ Lp (Ω) and we set p∗ = n−p
if n > p, p∗ = ∞ if
n = p.
2000 Mathematics Subject Classification. 35J60, 35B30, 35B40.
Key words and phrases. Elliptic systems; Nehari manifold; Local minimization; Ekeland variational principle.
c
2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted Jun 22, 2011. Published August 8, 2011.
1
2
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
Recently many authors have studied the existence of solutions for such problems
(see [2], [16], [10] and their references). The problem

 −∆p u = u|u|α−1 |v|β+1 + f
−∆q v = |u|α+1 v|v|β−1 + g

u=v=0
in Ω
in Ω
on ∂Ω
where Ω is a bounded domain, f ∈ D0−1,p0 (Ω), g ∈ D0−1,q0 (Ω) has been discussed by
β+1
chabrowski [7] with p = q and α+1
p∗ + q ∗ = 1, in [15] for p 6= q on bounded domain
and in a recent paper [4] on arbitrary domains with lack of compactness. In this
paper, we use the technique of J. Velin [15].
Let us define X = W01,p (Ω) × W01,q (Ω) equipped with the norm k(u, v)kX =
max(k∇ukp , k∇vkq ) and (X, k.k) is a reflexive and separable Banach Space.
Definition 1.1 (Weak Solution). We say that (u, v) ∈ X is a weak solution of
(1.1) if:
Z
|∇u|p−2 ∇u.∇w1 dx
Z
Z
Z
p−2
0
α−1
β+1
=λ
a(x)u|u| w1 dx + λ
c(x)u|u|
|v|
w1 dx +
f w1 dx,
Ω
Ω
Z
Ω
Ω
Ω
|∇v|q−2 ∇v.∇w2 dx
Z
Z
Z
q−2
0
α+1
β−1
=µ
b(x)v|v| w2 dx + λ
c(x)|u|
v|v|
w2 dx +
gw2 dx.
Ω
Ω
Ω
for all (w1 , w2 ) ∈ X.
Definition 1.2. We say that J satisfies the Palais-Smale condition (P S)c if every
sequence {(um , vm )} ⊂ X such that J(um , vm ) is bounded and J 0 (um , vm ) → 0 in
X ∗ as m → ∞ is relatively compact in X.
It is well known if J is bounded below and J has a minimizer on X, then this
minimizer is a critical point of J. However, the Euler function J(u, v), associated
with the problem (1.1), is not bounded below on the whole space X, but is bounded
on an appropriate subset, and has a minimizer on this set (if it exists) gives rise to
solution to (1.1).
Clearly, the critical points of J are the weak solutions of problem (1.1).
We set for all r > 0, t > 0 :
1
1
−
,
t
α+β+2
α+β+2−t
c(t) =
,
α+β+1
a(t) =
(r + 1)(t − 1)
,
λ0 (α + β + 2)(α + β + 1)
1
d(r, t) = α+1
.
β+1
0 + 0 q0
0
p
pr
q t
b(r, t) =
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEMS
3
and
p∗
q∗
p
h
θp ih b(α, p) min (spp , sqq ) i p∗ −p
ε1 = (α + 1)d(θ, γ) c(p) −
,
p
c0
p∗
q∗
q
h
γ q ih b(β, q) min (spp , sqq ) i q∗ −q
ε2 = (β + 1)d(θ, γ) c(q) −
,
q
c0
where θ, γ are fixed numbers such that
1
0 < θ < [pc(p)] p ,
1
0 < γ < [qc(q)] q .
The associated Euler-Lagrange functional to system (1.1) J : X → R is defined by
α+1
β+1
P (u) +
Q(v) − λ0 R(u, v) − (α + 1) < f, u > −(β + 1) < g, v >
p
q
(1.2)
Z
Z
where P (u) = k∇ukpp − λ
a(x)|u|p dx, Q(v) = k∇vkqq − µ
b(x)|v|q dx,
J(u, v) =
Ω
Z
Ω
c(x)|u|α+1 |v|β+1 dx.
R(u, v) =
Ω
Consider the Nehari manifold associated to problem (1.1) given by
Λ = {(u, v) ∈ X\{(0, 0)}; < J 0 (u, v), (u, v) >X ∗ ,X = 0}.
(1.3)
We define m1 = inf (u,v)∈Λ J(u, v).
Consequently, for every (u, v) in Λ, (1.2) becomes
J|Λ (u, v) = (α + 1)a(p)k∇ukpp + (β + 1)a(q)k∇vkqq − λa(p)(α + 1)
Z
a(x)|u|p dx
Ω
Z
− µ(β + 1)a(q)
b(x)|v|q dx − (α + 1)a(1) < f, u > −(β + 1)a(1) < g, v > . (1.4)
Ω
2. Results
0
0
Theorem 2.1. Suppose that f ∈ W0−1,p (Ω) and g ∈ W0−1,q (Ω) and Ω is a sufficiently regular bounded open set in Rn , and :
(a)
(c)
α+1 β+1
+
=1
(b) max (p, q) < α + β + 2
p∗
q∗
0 < kf k−1,p0 + kgk−1,q0 < min (1 , 2 , 1),
there exists a pair (u∗ , v ∗ ) ∈ Λ for the problem (1.1). Moreover, (u∗ , v ∗ ) satisfies
the property J(u∗ , v ∗ ) < 0.
Lemma 2.2. Suppose α + β + 2 > max(p, q). There exists a sequence (um , vm ) ∈ Λ
such that
1
inf J(u, v) < J(um , vm ) < inf J(u, v) +
,
(2.1)
m
(u,v)∈Λ
(u,v)∈Λ
and
1
0
kJ|Λ
(um , vm )kX ∗ ≤
.
(2.2)
m
4
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
Proof. We claim that J is bounded below on Λ. Let (u, v) be an arbitrary element
in Λ. Using successively the Holder, s inequality and the Young inequality on the
terms < f, u > and < g, v >, we can write
J|Λ (u, v) ≥ (α + 1)[a(p)k∇ukpp − θp k∇ukpp ] + (β + 1)[a(q)k∇ukqq − γ q k∇ukqq ]
0
0
0
0
−θ−p [a(1)kf k−1,p0 ]p − γ −q [a(1)kgk−1,q0 ]q .
This inequality follows from a(x), b(x) are sign chaining functions and we can choose
(u, v) ∈ X with these properties that supp u ⊆ Ω1 = {x ∈ Ω; a(x) < 0} and
supp v ⊆ Ω2 = {x ∈ Ω; b(x) < 0}.
Since the real numbers θ and γ being arbitrary, a suitable choose of θ and γ assure
that J is bounded below on Λ. The Ekeland variational principle ensures the
existence of such sequence.
Now, consider the function I defined on X by
I(u, v)
= < J 0 (u, v), (u, v) >
=
Z
(α + 1)k∇ukpp + (β + 1)k∇vkqq − λ(α + 1)
a(x)|u|p dx
Ω
Z
Z
q
0
−µ(β + 1)
b(x)|v| dx − λ (α + β + 2)
c(x)|u|α+1 |v|β+1 dx
Ω
Ω
−(α + 1) < f, u > −(β + 1) < g, v > .
(2.3)
We shall show that each minimizing sequence contains a Palais-Smale sequence
when f, g satisfied in
0 < kf k−1,p0 + kgk−1,q0 < min (1 , 2 , 1) .
0
(2.4)
∗
We want to establish that J (um , vm ) → 0 in X as m → ∞.
Lemma 2.3. The critical value of J on Λ, m1 = inf (u,v)∈Λ J(u, v), has the following property:
i
h α+1
0
0
β+1
m1 < min − 0 kf kp−1,p0 , − 0 kgkq−1,q0
p
q
Proof. Let uf be the unique solution of the Dirichlet problem
−∆p u = f
u=0
in Ω
on ∂Ω
and let vg be the unique solution of the problem
−∆q v = g
v=0
in Ω
on ∂Ω
It is clear that (uf , 0) , (0, vg ) are two elements of Λ and we have
1
m1 ≤ J(uf , 0) = (α + 1)[ k∇uf kpp − < f, uf >]
p
1
= −(α + 1)(1 − )k∇uf kpp
p
α+1
= − 0 k∇uf kpp ,
p
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEMS
1
m1 ≤ J(0, vg ) = (β + 1)[ k∇vg kqq − < g, vg >]
q
=
=
5
1
−(β + 1)(1 − )k∇vg kqq
q
β+1
− 0 k∇vg kqq ,
q
Similarly to proof of J. Velin [15, Lemma 4.2 ], we can show that
0
kf kp−1,p0 = k∇uf kpp ,
0
kgkq−1,q0 = k∇vg kqq ,
Then
m1 ≤ min
h
−
i
α+1
β+1
p0
q0
kf
k
,
−
kgk
0
0
−1,p
−1,q
p0
q0
Thus, the Lemma is proved.
Lemma 2.4. Under the condition (2.4), we have < I 0 (u, v), (u, v) >6= 0 for all
(u, v) ∈ Λ.
Proof. Suppose there exists some (û, v̂) in Λ such that I 0 (û, v̂) = 0. Then, from
Lemma 4.5 in [15], û and v̂ are not equal to zero. So (û, v̂) satisfies the obvious
relations
Z
Z
p
q
p
(α + 1)[kûk1,p − λ
a(x)|û| dx] + (β + 1)[kv̂k1,q − µ
b(x)|v̂|q dx]
Ω
−λ0 (α+β +2)
Z
Ω
c(x)|û|α+1 |v̂|β+1 dx−(α+1) < f, û > −(β +1) < g, v̂ >= 0, (2.5)
Ω
p(α + 1)[kûkp1,p − λ
Z
a(x)|û|p dx] + q(β + 1)[kv̂kq1,q − µ
Ω
−λ0 (α+β+2)2
Z
Z
b(x)|v̂|q dx]
Ω
c(x)|û|α+1 |v̂|β+1 dx−(α+1) < f, û > −(β+1) < g, v̂ >= 0. (2.6)
Ω
Combining (2.5) and (2.6), we obtain
Z
Z
b(x)|v̂|q dx]
(p − 1)(α + 1)[kûkp1,p − λ
a(x)|û|p dx] + (q − 1)(β + 1)[kv̂kq1,q − µ
Ω
Ω
= λ0 (α + β + 2)(α + β + 1)
Z
c(x)|û|α+1 |v̂|β+1 dx
(2.7)
Ω
Then (2.7) implies that there exists L > 0 depending only to α, β, p, q, n, Sp , Sq ,
such that
L < kûk1,p
or
L < kv̂k1,q .
Using successively the Holder, s inequality, the Young inequality and the Sobolev
inequalities
1
1
Spp kukp∗ ≤ kuk1,p
and
Sqq kvkq∗ ≤ kvk1,q ,
6
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
we have
Z
α+1
c(x)|û|
|v̂|
β+1
Z
≤ c(x)|û|α+1 |v̂|β+1 dx
Ω
Z
Z
β+1
α+1
q∗
p∗
≤ c0 ( |û|α+1× α+1 dx) p∗ ( |v̂|β+1× β+1 dx) q∗
dx
Ω
Ω
Ω
β+1
= c0 kûkα+1
p∗ kv̂kq ∗
∗
p∗
≤ c0
α+1
h kûkα+1×
∗
p
p∗
α+1
q
β+1× β+1
+
kv̂kq∗
i
q∗
β+1
i
β+1
q∗
kv̂k
∗
q
p∗
q∗
hα + 1
∗
∗ 1
1
β+1
≤ c0
× p∗ kûkp1,p +
× q∗ kv̂kq1,q .
∗
p∗
q
Spp
Sqq
= c0
hα + 1
∗
kûkpp∗ +
Then after dividing (2.7) by λ0 (α + β + 2)(α + β + 1), we obtain
Z
Z
h
i
h
b(α, p) kûkp1,p − λ
a(x)|û|p dx + b(β, q) kv̂kq1,q − µ
b(x)|v̂|q dx
Ω
Ω
≤ c0
hα + 1
p∗
1
×
p∗
∗
kûkp1,p
Spp
i
1
β+1
q∗
×
kv̂k
+
∗
1,q .
q
q∗
Sqq
Thus, on Ω1 and Ω2 we have
b(α, p)kûkp1,p + b(β, q)kv̂kq1,q ≤ c0
hα + 1
p∗
∗
×
1
p∗
p
∗
kûkp1,p +
Sp
i
∗
β+1
1
× q∗ kv̂kq1,q .
∗
q
Sqq
∗
To proceed further assume kv̂kq1,q ≤ kûkp1,p ( analogously, by a suitable adaptation
∗
∗
of this case, the final result is similar under the assumption kûkp1,p ≤ kv̂kq1,q ). So,
it follows that
∗
c0
α+1 β+1
kûkp1,p .
)
(2.8)
b(α, p)kûkp1,p ≤ ( ∗ +
p∗
q∗
p
q∗
min (Spp , Sqq )
p∗
Setting L =
h
q∗
b(α,p) min (Spp , Sq q )
c0
i p∗1−p
, we have
L ≤ kûk1,p .
(2.9)
We return to the identities (2.5) and (2.6). Multiply (2.5) by (α+β +2) and subtract
(2.6)from (α + β + 2) × (2.5). After some simplifications, we obtain
Z
Z
(α + 1)c(p)[kûkp1,p − λ
a(x)|û|p dx] + (β + 1)c(q)[kv̂kq1,q − µ
b(x)|v̂|q dx]
Ω
Ω
= (α + 1) < f, û > +(β + 1) < g, v̂ > .
(2.10)
,
Hence, using successively the Holder s inequality and the Young inequality and
properties of chaining sign functions a(x) and b(x) we have
h
h
i
i
q
p
(α + 1) c(p) − θp kûkp1,p + (β + 1) c(q) − γq kv̂kq1,q
≤ (α + 1)
0
0
1
1
kf kp−1,p0 + (β + 1) 0 q0 kgkq−1,q0 ,
p0 θp0
qγ
(2.11)
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEMS
7
where γ and µ denote arbitrary positive real numbers such that
1
0 < θ < [pc(p)] p
1
and
0 < γ < [qc(q)] q ,
From (2.11), we deduce
h
0
0
1
1
θp i
kûkp1,p ≤ (α + 1) 0 p0 kf kp−1,p0 + (β + 1) 0 q0 kgkq−1,q0
(α + 1) c(p) −
p
pθ
qγ
(2.12)
and
h
0
0
γq i
1
1
(β + 1) c(q) −
kv̂kq1,q ≤ (α + 1) 0 p0 kf kp−1,p0 + (β + 1) 0 q0 kgkq−1,q0 . (2.13)
q
pθ
qγ
From (2.9) and (2.12) becomes
p∗
q∗
p
0
α+1 β+1
θp ih b(α, p) min (Spp , Sqq ) i p∗ −p
≤ ( 0 p0 + 0 q0 )(kf kp−1,p0 +
(α + 1) c(p) −
p
c0
pθ
qγ
h
0
kgkq−1,q0 ).
Or more simply,
p∗
q∗
p
h
θp ih b(α, p) min (Spp , Sqq ) i p∗ −p
(α + 1)d(θ, γ) c(p) −
≤ kf k−1,p0 + kgk−1,q0 .
p
c0
(2.14)
p∗
q∗
With a similar argument, if we choose kûk1,p ≤ kv̂k1,q , we obtain
p∗
q∗
q
h
γ q ih b(β, q) min (Spp , Sqq ) i q∗ −q
≤ kf k−1,p0 + kgk−1,q0 .
(β + 1)d(θ, γ) c(q) −
q
c0
(2.15)
Consequently, we have
min (ε1 , ε2 ) < kf k−1,p0 + kgk−1,q0 .
This yields a contradiction with the assumption (2.4) and complete the proof.
1
1
Proposition 2.5. Let (θ, γ) ∈ R2 such that 0 < θ < [pc(p)] p , 0 < γ < [qc(q)] q .
0
0
Suppose that f ∈ W −1,p (Ω)\{0} and g ∈ W −1,q (Ω)\{0} satisfy the condition (8),
then there exists δ > 0 such that | < I 0 (um , vm ), (um , vm ) > | ≥ δ > 0.
Proof. Assume, for the sake of contradiction, that there exists a subsequence of
{(um , vm )} such that | < I 0 (um , vm ), (um , vm ) > | tends to 0as m → +∞. Then,
using the formula (2.3), we have
Z
Z
h
i
h
i
p(α + 1) kum kp1,p − λ
a(x)|um |p dx + q(β + 1) kvm kq1,q − µ
b(x)|vm |q dx −
Ω
(α+β+2)2 λ0
Z
Ω
c(x)|um |α+1 |vm |β+1 dx−(α+1) < f, um > −(β+1) < g, vm >= sm ,
Ω
(2.16)
where sm designate a real sequence tending to zero.
Moreover, as {(um , vm )} ⊂ Λ, we have also
Z
Z
h
i
h
i
p
q
p
(α + 1) kum k1,p − λ
a(x)|um | dx + (β + 1) kvm k1,q − µ
b(x)|vm |q dx −
Ω
Ω
8
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
(α + β + 2)λ0
Z
c(x)|um |α+1 |vm |β+1 dx − (α + 1) < f, um > −(β + 1) < g, vm >= 0.
Ω
(2.17)
Combining (2.16) and (2.17), we obtain
Z
h
i
h
p
(p − 1)(α + 1) kum k1,p − λ
a(x)|um |p dx + (q − 1)(β + 1) kvm kq1,q −
Ω
Z
µ
i
b(x)|vm |q dx = λ0 (α+β +2)(α+β +1)
Ω
Z
c(x)|um |α+1 |vm |β+1 dx+sm . (2.18)
Ω
∗
∗
We argue as in the proof of the Lemma 2.5. Suppose kum kp1,p ≤ kvm kq1,q . Similar
to (13), we obtain
α + 1 β + 1
∗
c0
+
kvm kq1,q−q + sm ,
(2.19)
b(β, q)kvm kq1,q ≤
p∗
q∗
p∗
q∗
p
q
min (Sp , Sq )
or, more simply,
p∗
q∗
∗
min (Spp , Sqq ) h
sm i
≤ kvm kq1,q−q .
b(β, q) −
c0
kvm kq1,q
(2.20)
We note that there exists a positive constant K such that kvm1k1,q ≤ K. In fact,
suppose the contrary. Then, kvm k1,q tends to 0 and kum k1,p also. We conclude that
J(um , vm ) tends to 0. But from (5) we deduce that m1 = 0, which is impossible
according to Lemma 2.3.
For m sufficiently large, we can assume
sm
< b(β, q).
kvm kq1,q
From this, we obtain the inequality
p∗
q∗
h b(β, q) min (S p , S q ) i q∗1−q
p
q
− Asm ≤ kvm k1,q ,
c0
where A is a constant depending only to α, β, p, q, p∗ , q ∗ , Sp , Sq .
We conclude the proof as in the closing stages of Lemma 2.4. We obtain for 0 <
1
1
θ < [pc(p)] p and 0 < γ < [qc(q)] q successively
p∗
q ∗ i ∗p
p −p
p
q
h
p ih b(α, p) min (Sp , Sq )
θ
(α + 1)d(θ, γ) c(p) −
− Asm ≤ kf k−1,p0 +
p
c0
kgk−1,q0 + sm
and
p∗
q∗
γ q b(β, q) min (Spp , Sqq ) q∗q−q
(β + 1)d(θ, γ)[c(q) − ][
]
− Bsm ≤ kf k−1,p0 +
q
c0
kgk−1,q0 + sm .
Letting m → ∞, we get
p∗
q∗
θp b(α, p) min (Spp , Sqq ) p∗p−p
(α + 1)d(θ, γ)[c(p) − ][
]
≤ kf k−1,p0 + kgk−1,q0
p
c0
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEMS
9
and
p∗
q∗
q
h
γ q ih b(β, q) min (Spp , Sqq ) i q∗ −q
(β + 1)d(θ, γ) c(q) −
≤ kf k−1,p0 + kgk−1,q0 .
q
c0
But this result contradicts the hypothesis (2.4) and the proof is complete.
Lemma 2.6. Let {(um , vm )} be a minimizing sequence for m1 = inf (u,v)∈Λ J(u, v).
Then, there exist u∗ ∈ W01,p (Ω), v ∗ ∈ W01,q (Ω) such that um * u∗ weakly in
W01,p (Ω) and vm * v ∗ weakly in W01,q (Ω).
Proof. We claim that {(um , vm )} is a bounded sequence of X. In fact, using (2.1)
and (2.2), we have
J(um , vm ) = m1 + om (1)
J 0 (um , vm ) = om (k(um , vm )kX ).
and
1
J 0 (um , vm )(um , vm )
α+β+2
Z
Z
= (α + 1)a(p)[kum kp1,p −
a(x)|um |p dx] + (β + 1)a(q)[kvm kq1,q −
b(x)|vm |q dx]
J(um , vm ) −
Ω
Ω
−(α + 1)a(1) < f, um > −(β + 1)a(1) < g, vm >
= m1 + om (k(um , vm )kX ) + om (1).
Using successively the Holder, s inequality, the Young inequality on the terms
< f, um > and < g, vm > and by Sobolev imbedding W 1,p (Ω) ,→ Lp (Ω) and
W 1,q (Ω) ,→ Lq (Ω) we can write
h
i
h
(α + 1) a(p)kum kp1,p − θp kum kp1,p − λc0 a(p)kum kp1,p + (β + 1) a(q)kvm kq1,q −
i
ν p kvm kq1,q − µc00 a(q)kvm kq1,q
0
0
0
0
≤ (α + 1)a(1)θ−p kf kp−1,p0 + (β + 1)a(1)ν −q kgkq−1,q0 + m1
+om (k(um , vm )kX ) + om (1).
Since the real numbers θ and ν being arbitrary, a suitable choose of θ and ν assure
that boundedness of the sequence {(um , vm )}.
We deduce that {(um , vm )} is a bounded sequence of X. We may extract two
subsequences denote again by {um } and {vm } converging weakly in W01,p (Ω) and
W01,q (Ω), respectively. Let u∗ and v ∗ be, respectively, the weak limits of {um } and
{vm }. (i, e :)
um * u∗
vm * v ∗
weakly
weakly
W01,p (Ω),
W01,q (Ω),
um → u∗
a.e. in Ω,
∗
a.e. in Ω.
vm → v
10
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
3. Proof of the Theorem 2.1
Taking again the minimizing sequence {(um , vm )}m∈N ⊂ Λ. We now show that
J 0 (um , vm ) → 0 in X ∗ as m → ∞. Since I 0 (u, v) 6= 0 on Λ, we have
0
J 0 (um , vm ) = J|Λ
(um , vm ) − λm I 0 (um , vm ),
for some λm ∈ R. Since {(um , vm )}m∈N ⊂ Λ, we have
0
0 =< J 0 (um , vm ), (um , vm ) >=< J|Λ
(um , vm ), (um , vm ) > −λm
< I 0 (um , vm ), (um , vm ) > .
Using Proposition 2.5, we conclude λm → 0 as m → ∞. Thus J 0 (um , vm ) → 0
0
in X ∗ , we see that Ju0 (um , vm ) → 0 and Jv0 (um , vm ) → 0 in W0−1,p (Ω). Consequently
−∆p um = λ a(x)um |um |p−2 + λ0 c(x)u|um |α−1 |vm |β+1 + f + fm
−∆q vm = µ b(x)vm |vm |q−2 + λ0 c(x)|um |α+1 vm |vm |β−1 + g + gm
0
in Ω
in Ω
0
with fm → 0 strongly in W −1,p (Ω)and gm → 0 strongly in W −1,q (Ω). Since wm =
0
λa(x)um |um |p−2 +λ0 c(x)um |um |α−1 |vm |β+1 ∈ W −1,p (Ω) and tm = µb(x)vm |vm |q−2
0
0
0
+λ0 c(x)|um |α+1 vm |vm |β−1 ∈ W −1,q (Ω) are bounded in W −1,p (Ω), W −1,q (Ω) respectively and in L1 (Ω), we can apply Theorem 2.1 from [5]. We obtain the strongly
convergence of ∇um to ∇u∗ in Lr (Ω)n for every r < p.
Similarly, we can show the strongly convergence ∇vm to ∇v ∗ in Ls (Ω)n for every
s < q.
From Remark 2.1.in [5] we have
|∇um |p−2 ∇um → |∇u∗ |p−2 ∇u∗
|∇um |p−2 ∇um * |∇u∗ |p−2 ∇u∗
∗
a.e. in
weakly in
Ω
0
(Lp (Ω))n .
(3.1)
(3.2)
∗
Proposition 3.1. The pair (u , v ) obtained in Lemma 2.6 is solution of the problem (1.1).
0
Proof. Let ψ ∈ W01,p (Ω) and ζ ∈ W01,q (Ω). For every (u, v) in X, we define J|u
and
0
J|v by
0
< J|u
(u, v), ψ >−1,1 =< J 0 (u, v), (ψ, 0) >X,X ∗
and
0
(u, v), ζ >−1,1 =< J 0 (u, v), (0, ζ) >X,X ∗ .
< J|v
Hence, taking u = um , v = vm , we have
Z
0
< J|u
(um , vm ), ψ >−1,1 =< −∆p um , ψ >−1,1 −λ
a(x)um |um |p−2 ψdx
Ω
Z
0
α−1
β+1
−λ
c(x)|um |
um |vm |
ψdx− < f, ψ >−1,1 − < fm , ψ >−1,1 ,
Ω
and
Z
0
< J|v
(um , vm ), ζ >−1,1 =< −∆q vm , ζ >−1,1 −µ
b(x)vm |vm |q−2 ζdx
Ω
Z
0
−λ
c(x)|um |α+1 vm |vm |β−1 ζdx− < g, ζ >−1,1 − < gm , ζ >−1,1
Ω
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEMS
11
passing to the limit on m from (3.2), we get
0
< J|u
(um , vm ), ψ >−1,1
Z
Z
=< −∆p u∗ , ψ >−1,1 −λ
a(x)u∗ |u∗ |p−2 ψdx − λ0
c(x)|u∗ |α−1 u∗ |v ∗ |β+1 ψdx
lim
m→+∞
Ω
Ω
− < f, ψ >−1,1
and
0
(um , vm ), ζ >−1,1
< J|v
Z
Z
∗
0
∗ ∗ q−2
0
=< −∆q v , ζ >−1,1 −λ
b(x)v |v | ζdx − λ
c(x)|u∗ |α+1 v ∗ |v ∗ |β−1 ψdx
lim
m→+∞
Ω
Ω
− < g, ζ >−1,1 .
Thus from (2.1), (2.2) we deduce for every ψ in W01,p (Ω)
Z
Z
< −∆p u∗ , ψ >−1,1 −λ
a(x)u∗ |u∗ |p−2 ψdx − λ0
c(x)|u∗ |α−1 u∗ |v ∗ |β+1 ψdx
Ω
Ω
− < f, ψ >−1,1 = 0
also, for every ζ in W01,q (Ω)
Z
Z
< −∆q v ∗ , ζ >−1,1 −λ0
b(x)v ∗ |v ∗ |q−2 ζdx − λ0
c(x)|u∗ |α+1 v ∗ |v ∗ |β−1 ψdx
Ω
Ω
− < g, ζ >−1,1 = 0
∗
∗
Therefore, (u , v ) is weak solution of (1.1).
On the other hand, we get
(a) < J 0 (u∗ , v ∗ ), (u∗ , v ∗ ) >−1,1 = 0,
(b) J(u∗ , v ∗ ) = m1 < 0
The result (a) shows that (u∗ , v ∗ ) ∈ Λ. Since (u∗ , v ∗ ) is the solution of (1.1), (a) is
obtained obviously by taking (ψ, ζ) = (u∗ , v ∗ ).
Now, we establish (b). Since m1 = inf (u,v)∈Λ J(u, v), (a) implies that m1 ≤ J(u∗ , v ∗ ).
1
, the weak semicontinuity of J|Λ
On the other hand, because J(um , vm ) < m1 + m
∗ ∗
ensures that J(u , v ) ≤ lim inf m→+∞ J(um , vm ) ≤ m1 . Then
m1 =
lim J(um , vm ) = J(u∗ , v ∗ )
m→+∞
By virtue of Lemma 2.3, we obtain J(u∗ , v ∗ ) < 0.
Proposition 3.2. There exist positive constants η1 , η2 such that for 0 < λ <
η1 , 0 < µ < η2 , the sequence {um } and {vm } converge strongly to u∗ and v ∗ in
W01,p (Ω) and W01,q (Ω), respectively.
Proof. Since limm→+∞ J 0 (um , vm ) = J(u∗ , v ∗ ) and (u∗ , v ∗ ) ∈ Λ, we write
Z
lim {(α + 1)a(p)[kum kp1,p − λ
a(x)|um |p dx] + (β + 1)a(q)[kvm kq1,q
m→+∞
Ω
Z
−µ
b(x)|vm |q dx] − (α + 1)a(1) < f, um > −(β + 1)a(1) < g, vm >}
Ω
12
G.A. AFROUZI, S. KHADEMLOO, M. MIRZAPOUR
Z
= (α + 1)a(p)[ku∗ kp1,p − λ
a(x)|u∗ |p dx] + (β + 1)a(q)[kv ∗ kq1,q
Ω
Z
−µ
b(x)|v ∗ |q dx] − (α + 1)a(1) < f, u∗ > −(β + 1)a(1) < g, v ∗ > .
Ω
Because limm→+∞ < f, um >=< f, u∗ > and limm→+∞ < g, vm >=< g, v ∗ >,
we deduce
Z
p
lim {(α + 1)a(p)[kum k1,p − λ
a(x)|um |p dx] + (β + 1)a(q)[kvm kq1,q
m→+∞
Ω
Z
q
−µ
b(x)|vm | dx]}
Ω
Z
∗ p
= (α + 1)a(p)[ku k1,p − λ
a(x)|u∗ |p dx] + (β + 1)a(q)
Ω
Z
∗ q
[kv k1,q − µ
b(x)|v ∗ |q dx]
(3.3)
Ω
Which implies that (um , vm ) → (u∗ , v ∗ ) strongly in X. Suppose the contrary.
Then, um being weakly convergent in W01,p (Ω), we may assume that there exists µp
∗
and γp two measure such that |∇um |p converges weak ∗ to µp and |um |P converges
weak∗ to γp .
According to the concentration-compactness principle due to Lions [13], there exists
an at most countable index set Γ, positive constants {γpj } , {µpj } (j ∈ Γ) and
collection of points {xj }j∈Γ in Ω such that, for all j ∈ Γ
∗
γp = |u∗ |p +
X
γpj δxj
(3.4)
j∈Γ
µp ≥ |∇u∗ |p +
X
µpj δxj
(3.5)
j∈Γ
p
∗
γppj ≤
µpj
.
Sp
(3.6)
Integrating (3.5) over Ω, we obtain
Z
Z
X
p
lim
|∇um | dx ≥
|∇u∗ |p dx +
µpj ({xj }).
m→+∞
Ω
Ω
Similarly, by the same arguments, we also obtain
Z
Z
X
lim
|∇vm |q dx ≥
|∇v ∗ |q dx +
µql ({xj }).
m→+∞
Ω
(3.7)
j∈Γ
Ω
(3.8)
l∈Γ
By Sobolev imbedding W 1,p (Ω) ,→ Lp (Ω and W 1,q (Ω) ,→ Lq (Ω) there exist positive
constants c0 and c00 such that
Z
Z
p
p
0
a(x)|um | dx ≤ c kum k1,p and
b(x)|vm |q dx ≤ c00 kvm kq1,q ,
Ω
Ω
ON POSITIVE SOLUTION FOR A CLASS OF NONLINEAR ELLIPTIC SYSTEMS
13
then from (3.3) we get
lim {(α + 1)a(p)[kum kp1,p − λ
m→+∞
Z
−µ
b(x)|vm |q dx]}
Z
a(x)|um |p dx] + (β + 1)a(q)[kvm kq1,q
Ω
Ω
≥
lim {(α + 1)a(p)[kum kp1,p − λ c0 kum kp1,p ] + (β + 1)a(q)[kvm kq1,q
m→+∞
00
−µ c kvm kq1,q ]}
= {(α + 1)a(p)(1 − λ c0 )kum kp1,p + (β + 1)a(q)(1 − µ c00 )kvm kq1,q }
Let η1 = c10 and η2 = c100 . If we multiply (3.7) by (α + 1)a(p)(1 − λc0 ), (3.8) by
(β + 1)a(q)(1 − µc00 ) we obtain
{(α + 1)a(p)(1 − λ c0 )kum kp1,p + (β + 1)a(q)(1 − µ c00 )kvm kq1,q }
X
≥ (α + 1)a(p)(1 − λc0 )ku∗ kp1,p + (α + 1)a(p)(1 − λc0 )
µpj ({xj })
j∈Γ
00
+(β + 1)a(q)(1 − µc
)kvm kq1,q
00
+ (β + 1)a(q)(1 − µc )
X
µql ({xj })
l∈Γ
Z
Z
≥ (α + 1)a(p)[ku∗ kp1,p − λ
|u∗ |p dx] + (β + 1)a(q)[kv ∗ kq1,q − µ
b(x)|v ∗ |q dx]
Ω
Ω
X
X
+(α + 1)a(p)(1 − λc0 )
µpj ({xj }) + (β + 1)a(q)(1 − µc00 )
µql ({xj }).
j∈Γ
l∈Γ
Then, from (3.3) we deduce
X
X
(α + 1)a(p)(1 − λ c0 )
µpj ({xj }) + (β + 1)a(q)(1 − µ c00 )
µql ({xj }) ≤ 0 .
j∈Γ
l∈Γ
This is impossible and the proof is complete.
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
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Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Basic Sciences,, University of Mazandaran,
Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Somayeh Khademloo
Babol (Noushirvani) University of Technology, Iran
E-mail address: S.khademloo@nit.ac.ir
Maryam Mirzapour
Department of Mathematics, Faculty of Basic Sciences,, University of Mazandaran,
Babolsar, Iran
E-mail address: mirzapour@stu.umz.ac.ir