Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 142, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR QUASILINEAR ELLIPTIC
DEGENERATE SYSTEMS WITH L1 DATA AND
NONLINEARITY IN THE GRADIENT
ABDELHAQ MOUIDA, NOUREDDINE ALAA, SALIM MESBAHI, WALID BOUARIFI
Abstract. In this article we show the existence of weak solutions for some
quasilinear degenerate elliptic systems arising in modeling chemotaxis and angiogenesis. The nonlinearity we consider has critical growth with respect to
the gradient and the data are in L1 .
1. Introduction
Reaction-diffusion systems are important for a wide range of applied areas such
as cell processes, drug release, ecology, spread of diseases, industrial catalytic processes, transport of contaminants in the environment, chemistry in interstellar media, to mention a few. Some of these applications, especially in chemistry and
biology, are explained in books by Murray [26, 27] and Baker [10]. While a general
theory of reaction-diffusion systems is detailed in the books of Rothe [34] and Grzybowski [21]. Various forms of this problems have been proposed in the literature.
Most discussions in the current literature are for linear or nonlinear systems and
different methods for the existence problem have been used, see Alaa et al [1]–[9],
Baras [11, 12], Boccardo et al [15], Boudiba [16] and Pierre et al [29]-[32]. This is
a relatively recent subject of mathematical and applied research. Most of the work
that has been done so far is concerned with the exploration of particular aspects
of very specific systems and equations. This is because these systems are usually
very complex and display a wide range of phenomena remain poorly understood.
Consequently, there is no established program for solving a large class of systems.
For example a system of Chemotaxis, which is a biological phenomenon describing the change of motion of a population densities or of single particles (such as
amoebae, bacteria, endothelial cells, any cell, animals, etc.) in response (taxis) to
an external chemical stimulus spread in the environment where they reside see for
example [28]. The simple mathematical model which describes such a phenomenon
2000 Mathematics Subject Classification. 35J55, 35J60, 35J70.
Key words and phrases. Degenerate elliptic systems; auasilinear; chemotaxis;
angiogenesis; weak solutions.
c
2013
Texas State University - San Marcos.
Submitted December 7, 2012. Published June 22, 2013.
1
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A. MOUIDA, N. ALAA, S. MESBAHI, W. BOUARIFI
EJDE-2013/142
reads as follows
∂u
− Du ∆u + ∇(κ(u)∇u + χ(v)∇v) = 0 in Ω × (0, T )
∂t
∂v
− Dv ∆v + ∇(ζ(u)∇u + η(v)∇v) = 0 in Ω × (0, T )
∂t
u(0) = u0 , v0 (0) = v0
(1.1)
here u and v are the population densities. For a simple expansion, we include
∂κ(u)
∂χ(v)
∂ζ(u)
∂η(v)
a(x) =
, b(x) =
, c(x) =
, d(x) =
∂u
∂v
∂u
∂v
f = −(κ(u)∆u + χ(v)∆)v, g = −(ζ(u)∆u + η(v)∆v) .
Then the system can be written as
∂u
− Du ∆u + a(x)|∇u|2 + b(x)|∇v|2 = f in Ω × (0, T )
∂t
∂v
− Dv ∆v + c(x)|∇u|2 + d(x)|∇v|2 = g in Ω × (0, T )
∂t
u(0) = u0 , v0 (0) = v0 .
(1.2)
In this work we are interested in the quasilinear elliplic degenerate problem
u − D1 ∆u + a(x)|∇u|2 + b(x)|∇v|α = f (x)
in Ω
v − D2 ∆v + c(x)|∇u|β + d(x)|∇v|2 = g(x)
in Ω
u=v=0
(1.3)
on ∂Ω
where Ω is an open bounded set of RN , N ≥ 1, with smooth boundary ∂Ω, the
diffusion coefficients D1 and D2 are positive constants, a, b, c, d, f, g : Ω → [0, +∞)
are a non-negative integrable functions and 1 ≤ α, β ≤ 2.
We are interested in the case where the data are non-regular and where the
growth of the nonlinear terms is arbitrary with respect to the gradient. To help
understanding the situation, let us mention some previous works concerning the
problem when a, b, c, d ∈ L∞ (Ω).
• if f, g are regular enough (f, g ∈ W 1,∞ (Ω)) and for all α, β ≥ 1, the method of
sub- and super-solution can be used to prove the existence of solutions to (1.3). For
instance (0, 0) is a subsolution and a solution, w = (w1 , w2 ), of the linear problem
w1 − D1 ∆w2 = f (x)
in Ω
w1 − D1 ∆w2 = g(x)
in Ω
w1 = w2 = 0
(1.4)
on ∂Ω,
is a supersolution. Then (1.3) has a solution (u, v) ∈ W01,∞ (Ω) ∩ W 2,p (Ω); see Lions
[23].
• If f, g ∈ L2 (Ω) and 1 ≤ α, β ≤ 2, then |∇u|α , |∇v|β ∈ L1 (Ω). Many authors
have studied this problem and showed that (1.3) has a solution (u, v) ∈ H01 (Ω) ×
H01 (Ω), see Bensoussan et al [14], Boccardo et al [15] and the references there in.
• If f, g ∈ L1 (Ω) and 1 ≤ α, β < 2, Alaa and Mesbahi [1] proved that (1.3) has
a non negative solution (u, v) ∈ W01,1 (Ω) × W01,1 (Ω).
• The case where f, g ∈ MB+ (Ω) (f, g are a finite non negative measures on Ω)
has treated by Alaa and Pierre [9]. They proved that if 1 ≤ α, β ≤ 2 and the
supersolution w = (w1 , w2 ) ∈ H01 (Ω) × H01 (Ω), then the problem (1.3) has a non
negative solution (u, v) ∈ H01 (Ω) × H01 (Ω).
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EXISTENCE OF SOLUTIONS
3
We are particularly interested in the case of a system (1.3) when a, b, c, d, f, g
are not regular, more precisely, a, b, c, d, f, g are in L1 (Ω).
Let us make some specifications on the model problem
u − D1 ∆u + b(r)|∇v|α = f
β
v − D2 ∆v + c(r)|∇u| = g
u=v=0
in B
in B
(1.5)
if N = 2
if N ≥ 3 .
(1.6)
on ∂B
where B is the unit ball in RN , r = kxk and
(
− ln r
b(r) = c(r) =
r2−N
In this case, b(r), c(r) are in L1loc (B) but not in L∞ (B). As a consequence the
techniques usually used to prove existence and based on a priori L∞ -estimates on
u and ∇u fail. To overcome this difficulty, we will develop a new method which
differ completely of the previous approach.
We have organized this article as follows. In section 2 we give the precise setting
of the problem and state the main result. In section 3 we present an approximate
problem and we give suitable estimates to prove that (1.3) has a solution in the
case where the growth of the nonlinearity with respect to the gradient is arbitrary.
2. Assumptions and statement of main results
Let f, g, a, b, c, d are functions that satisfies the following assumptions
f, g ∈ L1 (Ω),
a, b, c, d ∈
f, g ≥ 0
L1loc (Ω),
(2.1)
a, b, c, d ≥ 0
(2.2)
First, we have to clarify in which sense we want to solved problem (1.3).
Definition 2.1. We say that (u, v) is a weak solution of (1.3) if
u, v ∈ W01,1 (Ω)
a(x)|∇u|2 ,
b(x)|∇v|α ,
c(x)|∇u|β ,
d(x)|∇v|2 ∈ L1loc (Ω)
u − D1 ∆u + a(x)|∇u|2 + b(x)|∇v|α = f (x)
in D0 (Ω)
v − D2 ∆v + c(x)|∇u|β + d(x)|∇v|2 = g(x)
in D0 (Ω)
(2.3)
We are interested to proving the existence of weak positive solutions of the
problem (1.3). For this, we define the truncation function Tk ∈ C 2 , such that
Tk (r) = r
if 0 ≤ r ≤ k
Tk (r) ≤ k + 1
if r ≥ k
Tk0 (r)
if r ≥ 0
0≤
Tk0 (r)
0≤
=0
≤1
if r ≥ k + 1
−Tk00 (r)
≤ C(k) .
(2.4)
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A. MOUIDA, N. ALAA, S. MESBAHI, W. BOUARIFI
EJDE-2013/142
For example, the function Tk can be defined as
Tk (r) = r in [0, k]
1
Tk (r) = (r − k)4 − (r − k)3 + r in [k, k + 1]
(2.5)
2
1
Tk (r) = (k + 1) for r > k + 1 .
2
Then we define the the space
τ 1,2 (Ω) = w : Ω → R measurable, such that Tk (w) ∈ H 1 (Ω) for all k > 0
This enables us to state the main result of this paper.
Theorem 2.2. Assume that (2.1) and (2.2) hold, and 1 ≤ α, β < 2. If there exists
a function θ ∈ τ 1,2 (Ω) and a sequence θn ∈ L∞ (Ω) such that
0 ≤ a, b, c, d ≤ θ in Ω
θn → θ
a.e. Ω
∇Tk (θn ) → ∇Tk (θ) strongly in L2 (Ω)
1 Z
lim sup
|∇Tk (θn )|2 = 0
k→∞ n
k Ω
Then the problem (1.3) has a non negative weak solution.
(2.6)
Remark 2.3. (i) If a, b, c, d ∈ L∞ (Ω), then (2.6) is satisfied. Indeed, θ can take
the value of any non negative constant C, such that
C ≥ max kakL∞ , kbkL∞ , kckL∞ , kdkL∞
(2.7)
(ii) Hypothesis (2.6) holds for the functions ξ = b or c given in (1.6). Indeed
−∆ξ = λ is in this case the measure of Dirac which is a finite non negative measure
on Ω. By consequent, we take θ = ξ and θn solution of
−∆θn = λn in Ω
(2.8)
θn = 0 on ∂Ω,
where λn ∈ C0∞ (Ω), λn → λ in L1 (Ω) and λn ≤ λ. Then, we can applied Theorem
2.2 and conclude the existence of the non negative weak solution for our model
problem (2.1).
3. Proof of theorem 2.2
3.1. An approximation scheme. In this paragraph, we define an approximated
system of (1.3). For this, we truncate the functions a, b, c, d, f, g by introducing the
sequence an , bn , cn , dn , fn , gn defined as follows
an = min{a, θn },
and
bn = min{b, θn },
cn = min{c, θn },
dn = min{d, θn }
fn ∈ C0∞ (Ω), fn → f in L1 (Ω), fn ≤ f
(3.1)
gn ∈ C0∞ (Ω), gn → g in L1 (Ω), gn ≤ g
Then the approximate problem is
un , vn ∈ W01,∞ (Ω)
un − D1 ∆un + an (x)|∇un |2 + bn (x)|∇vn |α = fn (x)
β
2
vn − D2 ∆vn + cn (x)|∇un | + dn (x)|∇vn | = gn (x)
in D0 (Ω)
0
in D (Ω) .
(3.2)
EJDE-2013/142
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5
One can see that an , bn , cn , dn are in L∞ (Ω). On the other hand, (0, 0) is a subsolution of (3.2) and (Un , Vn ) a solution of the linear problem
Un − D1 ∆Un = fn
in Ω
Vn − D2 ∆Vn = gn
in Ω
Un , Vn ∈
(3.3)
W01,∞ (Ω)
is a supersolution, then by the classical results in Amann and Grandall [13] and
Lions [23, 24], there exists (un , vn ) solution of (3.2) such that
0 ≤ un ≤ Un
for all n
0 ≤ vn ≤ Vn
for all n
3.2. A priori estimates. To prove theorem 2.2, we propose to send n to infinity
in (3.2). For this we will need some estimates passing to the limit.
Lemma 3.1. Let un , vn , an , bn , cn , dn be sequences defined as above. Then (i)
Z
|∇Tk (un )|2 ≤ kkf kL1 (Ω)
ZΩ
|∇Tk (vn )|2 ≤ kkgkL1 (Ω)
Ω
and (ii)
Z
bn .|∇Tk (vn )|α ≤ kkf kL1 (Ω)
Ω
Z
cn .|∇Tk (un )|β ≤ kkgkL1 (Ω)
Ω
Proof. (i) By multiplying the first equation of (3.2) by Tk (un ) and the second
equation by Tk (vn ) and integrating over Ω, we have
Z
Z
|Tk (un )|2 + D1
|∇Tk (un )|2
Ω
Ω
Z
Z
Z
2
α
+
an Tk (un )|∇Tk (un )| +
bn Tk (un )|∇Tk (vn )| ≤
fn Tk (un )
Ω
Ω
Ω
and
Z
Z
|Tk (vn )|2 + D2
|∇Tk (vn )|2
Ω
Ω
Z
Z
Z
β
2
+
cn Tk (vn )|∇Tk (un )| +
dn Tk (vn )|∇Tk (vn )| ≤
gn Tk (vn )
Ω
Ω
Ω
Thanks to the positivity of an , bn , cn , dn , the assumptions on fn and gn , the definition of the function Tk , we deduce the result.
(ii) Integrating the first equation of (3.2) over Ω, we obtain
Z
Z
Z
Z
Z
2
α
un − D1
∆un +
an (x)|∇un | +
bn (x)|∇vn | =
fn (x)
(3.4)
Ω
Ω
Ω
Ω
Ω
On the other hand, it is well know that for every function y in W01,1 (Ω) such that
−∆y = H, H ∈ L1 (Ω)
y≥0
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A. MOUIDA, N. ALAA, S. MESBAHI, W. BOUARIFI
EJDE-2013/142
there exists a sequence yn in C 2 (Ω) ∩ C0 (Ω) which satisfies
yn → y
strongly in W01,1 (Ω)
strongly in L1 (Ω)
∆yn → ∆y
The regularity of yn allows us to write
Z
Z
∆yn =
Ω
∂Ω
∂yn
dσ,
∂υ
n
but yn ≥ 0R on Ω and yn = 0 in ∂Ω. Then ∂y
∂υ ≤ 0. We deduce by passing to the
limit that Ω ∆y ≤ 0. Therefore
Z
∆un ≤ 0
Ω
The relation (3.4) yields
Z
Z
Z
Z
un +
an (x)|∇un |2 +
bn (x)|∇vn |α ≤
fn (x) .
Ω
Ω
Ω
Ω
By (3.1); we conclude that
Z
Z
Z
2
un +
an (x)|∇un | +
bn (x)|∇vn |α ≤ kf kL1 (Ω) .
Ω
Ω
Ω
In the same way, if we integrate the second equation of (3.2) over Ω, we obtain
Z
Z
Z
β
vn +
cn (x)|∇un | +
dn (x)|∇vn |2 ≤ kgkL1 (Ω) ,
Ω
Ω
Ω
hence the result follows.
Remark 3.2. (1) Using the assertion (ii) of lemma 3.1, and the compactness of
the operator
L1 (Ω) → W01,q (Ω)
G 7→ ϑ
where 1 ≤ q <
N
N −1 ,
and ϑ is the solution of the problem
ϑ ∈ W01,q (Ω)
αϑ − ∆ϑ = G in D0 (Ω)
we conclude the existence of u, up to a subsequence, still denoted by un for simplicity, such that
N
,
un → u strongly in W01,q (Ω), 1 ≤ q <
N −1
(un , ∇un ) → (u, ∇u) a.e. in Ω
see Brezis [17]
(2) Assertion (i) implies that
(Tk (un ), Tk (vn )) → (Tk (u), Tk (v))
weakly in H01 (Ω) × H01 (Ω)
Lemma 3.3. Let (un , vn ) be a solution of (3.2), then
1 Z
1 Z
2
lim sup
|∇Th (un )| dx = lim sup
|∇Th (vn )|2 dx = 0
h→+∞ n
h→+∞ n
h Ω
h Ω
EJDE-2013/142
EXISTENCE OF SOLUTIONS
7
Proof. We first remark that un satisfies
in D0 (Ω)
−∆un ≤ fn
If we multiply this inequality by Th (un ) and integrate on Ω, we obtain for every
0 < M < h,
Z
Z
Z
2
|∇Th (un )| ≤
f Th (un ) +
f Th (un )
Ω
Ω∩{un ≤M }
Ω∩{un >M }
Z
Z
≤M
f +h
f χ{un >M }
Ω
Ω
hence
Z
Z
M
|∇Th (un )| ≤
f+
f χ{un >M }
h Ω
Ω
Ω
Z
C
1
kun kL1 ≤
|{un > M }| =
dx ≤
M
M
{un >M }
Then limM →+∞ supn |{un > M }| = 0
On other hand, since f ∈ L1 (Ω), we have for each ε > 0 there exists δ such that
for for all E ⊂ Ω,
Z
ε
|E| < δ
|f | ≤ .
2
E
Taking into account the above limit, we obtain that for each ε > 0, there exists Mε
such that for all M ≥ Mε ,
Z
ε
sup
f χ[un >M ] ≤
2
n
Ω
1
h
Z
2
Taking M = Mε and letting h tend to infinity, we obtain
1 Z
lim sup
|∇Th (un )|2 = 0 .
h→∞ n
h Ω
Lemma 3.4. Let ηn be sequence such that ηn → η, a.e. in Ω and
then ηn → η in Lα (Ω) for all 1 ≤ α < 2.
R
Ω
|ηn |2 ≤ C
Proof. We show that ηn is equi-integrable in Lα (Ω). Let E be a measurable subset
of Ω; we have
Z
Z
α/2
α
(2−α)/2
|ηn | ≤ |E|
|ηn |2
≤ C|E|(2−α)/2
E
E
1 ≤ α < 2 then 0 < 2 − α ≤ 1. We choose |E| = ( Cε )2/(2−α) , we obtain
R Since
α
|η | ≤ ε.
E n
3.3. Convergence. The aim of this paragraph is to prove that (u, v) (obtained in
the previous section) is in fact a solution of problem (1.3). According to definition
2.1, we have to show only that
u − D1 ∆u + a(x)|∇u|2 + b(x)|∇v|α = f (x)
in D0 (Ω)
v − D2 ∆v + c(x)|∇u|β + d(x)|∇v|2 = g(x)
in D0 (Ω)
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A. MOUIDA, N. ALAA, S. MESBAHI, W. BOUARIFI
EJDE-2013/142
By lemma 3.1, we know that an (x)|∇un |2 , dn (x)|∇vn |2 , bn (x)|∇vn |α , cn (x)|∇un |β
are uniformly bounded in L1 (Ω). Moreover
an (x)|∇un |2 ≥ 0,
dn (x)|∇un |2 ≥ 0,
bn (x)|∇un |α ≥ 0,
cn (x)|∇un |β ≥ 0
and for almost every x in Ω, we have
an (x)|∇un (x)|2 → a(x)|∇u(x)|2
dn (x)|∇vn (x)|2 → d(x)|∇v(x)|2
bn (x)|∇vn (x)|α → b(x)|∇v(x)|α
cn (x)|∇un (x)|β → c(x)|∇u(x)|β
Then there exists µ1 , µ2 non negative measures, see Schwartz [35], such that
lim (un − D1 ∆un + an (x)|∇un |2 + bn (x)|∇vn |α )
n→+∞
in D0 (Ω)
= u − D1 ∆u + a(x)|∇u|2 + b(x)|∇v|α + µ1
lim (vn − D2 ∆vn + cn (x)|∇un |β + dn (x)|∇vn |2 )
n→+∞
in D0 (Ω)
= v − D2 ∆v + c(x)|∇u|β + d(x)|∇v|2 + µ2
Consequently,
u − D1 ∆u + a(x)|∇u|2 + b(x)|∇v|α ≤ f
in D0 (Ω)
v − D2 ∆v + c(x)|∇u|β + d(x)|∇v|2 ≤ g
in D0 (Ω)
Therefore, to conclude the proof of Theorem 2.2, we must establish the opposite
inequality. For this, Let H be a function in C 1 (R), such that
0 ≤ H(s) ≤ 1
(
0 if |s| ≥ 1
H(s) =
1 if |s| ≤ 21
To this end, we introduce the test functions
θn
un
θn
un ]H( )H( ),
D1
k
k
θn
θn
vn
Φ2 = ψ2 exp[−
vn ]H( )H( )
D2
k
k
Φ1 = ψ1 exp[−
where H denotes the function defined above and ψ1 , ψ2 ≤ 0, ψ1 , ψ2 ∈ H01 (Ω) ∩
L∞ (Ω). We multiply the first equation in (3.2) by Φ1 and we integrate on Ω, we
obtain
Z
X
fn Φ1 =
Ij ,
Ω
1≤j≤7
where
Z
I1 =
un Φ1 ,
Ω
Z
I2 = D1
θn
un
θn
∇un ∇ψ1 exp[−
un ]H( )H( )
D1
k
k
Ω
Z
I3 = −
un ∇un ∇θn Φ1
Ω
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EXISTENCE OF SOLUTIONS
D1
k
Z
θn
θn
un
∇un ∇θn ψ1 exp[−
un ]H 0 ( )H( )
D1
k
k
Ω
Z
I5 = (an − θn )|∇un |2 Φ1
Ω
Z
θn
D1
θn
un
|∇un |2 ψ1 exp[−
un ]H( )H 0 ( )
I6 =
k Ω
D1
k
k
Z
I7 =
bn .|∇vn |α Φ1
I4 =
Ω
Next we study each term. For the first term, we have
Z
θn
un
θn
un ]H( )H( )
lim I1 = lim
Tk (un )ψ1 exp[−
n→+∞
n→+∞ Ω
D1
k
k
Z
θ
u
θ
u]H( )H( )
=
uψ1 exp[−
D1
k
k
Ω
since
ψ1 exp[−
θn
un
θn
un ]H( )H( )
D1
k
k
converges strongly in L2 (Ω) to
ψ1 exp[−
θ
u
θ
u]H( )H( )
D1
k
k
in L2 (Ω)
and ∇Tk (un ) converges weakly to ∇Tk (u) in L2 (Ω), (see [24, lemma 1.3, p 12]).
Concerning the second term, we get
Z
θn
un
θn
un ]H( )H( )
lim I2 = lim D1
∇Tk (un )∇ψ1 exp[−
n→+∞
n→+∞
D1
k
k
Ω
Z
u
θ
θ
= D1
∇u∇ψ1 exp[−
u]H( )H( )
D1
k
k
Ω
since
∇ψ1 exp[−
θn
un
θn
un ]H( )H( )
D1
k
k
converges strongly in L2 (Ω) to
∇ψ1 exp[−
θ
u
θ
u]H( )H( ) .
D1
k
k
For I3 , we first remark that
Z
θn
un
θn
lim I3 = − lim
Tk (un )∇Tk (un )∇Tk (θn )ψ1 exp[−
un ]H( )H( )
n→+∞
n→+∞ Ω
D1
k
k
Z
θ
θ
u
=−
Tk (u)∇Tk (u)∇Tk (θ)ψ1 exp[−
u]H( )H( )
D1
k
k
Ω
since
Tk (un ) → Tk (u)
weakly in H01 (Ω)
Tk (θn ) → Tk (θ)
strongly in H01 (Ω)
To study I4 and I6 we use Lemma 3.3. For I4 , we have
h1 Z
θn
θn
un i1/2
I4 ≤ D 1
|∇Tk (un )|2 ψ1 exp[−
un ]H 0 ( )H( )
k Ω
D1
k
k
9
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A. MOUIDA, N. ALAA, S. MESBAHI, W. BOUARIFI
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k
EJDE-2013/142
Z
θn
θn
un
|∇Tk (θn )|2 ψ1 exp[−
un ]H 0 ( )H( )]1/2
D1
k
k
Ω
Z
Z
1
1
2 1/2
≤ D1 [kψ1 kL∞ (Ω)
|∇Tk (un )| ] [kψ1 kL∞ (Ω)
|∇Tk (θn )|2 ]12
k Ω
k Ω
×[
θn
un ] ≤ 1, thus
since exp[− D
1
I4 ≤ D1 [kψ1 kL∞ δk ]1/2 [kψ1 kL∞ ρk ]1/2
Where
Z
Z
1
1
|∇Tk (un )|2 ) and ρk = sup(
|∇Tk (θn )|2 )
n k Ω
n k Ω
By Lemma 3.3, we have
δk = sup(
lim δk = 0,
k→∞
lim ρk = 0
k→∞
Then
lim sup(I4 ) = 0
k→∞ n
Similarly, for I6 , we have
I6 ≤ D1 kψ1 kL∞ δk
Then
lim sup(I6 ) = 0
k→∞ n
Now we investigate the remaining term I5 . Since an ≤ θn and ψ1 ≤ 0, we have
θn
un
θn
u]H( )H( ) ≥ 0
in Ω
(an − θn )|∇un |2 exp[−
D1
k
k
Therefore, by Fatou’s lemma, we obtain
Z
u
θ
θ
lim I5 ≥ (a − θ)|∇u|2 exp[−
u]H( )H( )
n→+∞
D1
k
k
Ω
For I7 , we obtain
Z
θn
un
θn
lim I7 = lim
Tk (bn )|∇Tk (vn )|α ψ1 exp[−
un ]H( )H( )
n→+∞
n→+∞ Ω
D1
k
k
By a direct application of Lemma 3.3, we have |∇Tk (vn )|α → |∇Tk (v)|α strongly in
L1 (Ω), then
Z
θ
θ
u
lim I7 =
b|∇v|α ψ1 exp[−
u]H( )H( )
n→+∞
D
k
k
1
Ω
We have shown that
Z
Z
θ
θ
u
θ
θ
u
1
uψ1 exp[−
u]H( )H( ) + D1
∇u∇ψ1 exp[−
u]H( )H( )
ω( ) +
k
D1
k
k
D1
k
k
Ω
Z
Z Ω
θ
θ
u
θ
θ
u
−
u∇u∇θψ1 exp[−
u]H( )H( ) + (a − θ)|∇u|2 exp[−
u]H( )H( )
D
k
k
D
k
k
1
1
Ω
ZΩ
θ
θ
u
+
b|∇v|α ψ1 exp[−
u]H( )H( )
D
k
k
1
ZΩ
θ
θ
u
≤
f ψ1 exp[−
u]H( )H( )
D1
k
k
Ω
where ω(ε) denotes a quantity that tends to 0 when ε tends to 0. Now we choose
θ
θ
u
ψ1 = −ϕ1 exp[
u]H( )H( )
D1
k
k
EJDE-2013/142
EXISTENCE OF SOLUTIONS
11
where ϕ1 ≥ 0, ϕ1 ∈ D(Ω) and we replace ψ1 by this value in the previous inequality
to obtain
Z
Z
1
θ
u
u
θ
w( ) −
uϕ1 H 2 ( )H 2 ( ) − D1
∇u∇ϕ1 H 2 ( )H 2 ( )
k
k
k
k
k
Ω
Z
Z Ω
u
u
θ
θ
−
ϕ1 u∇u∇θH 2 ( )H 2 ( ) −
ϕ1 |∇u|2 θH 2 ( )H 2 ( )
k
k
k
k
Ω
Ω
Z
Z
D1
θ
θ
u
D
θ
u
u
1
−
ϕ1 ∇u∇θH 0 ( )H( )H 2 ( ) −
ϕ1 |∇u|2 H 2 ( )H( )H 0 ( )
k Ω
k
k
k
k Ω
k
k
k
Z
Z
2 u
2 u
2 θ
α
2 θ
+
b|∇v| ϕ1 H ( )H ( )
u∇u∇θϕ1 H ( )H ( ) −
k
k
k
k
ZΩ
ZΩ
θ
u
u
θ
−
a|∇u|2 ϕ1 H 2 ( )H 2 ( ) +
θϕ1 |∇u|2 H 2 ( )H 2 ( )
k
k
k
k
Ω
Ω
Z
u
θ
≤−
f ϕ1 H 2 ( )H 2 ( )
k
k
Ω
By developing calculations and remarking that the sixth and seventh terms are
equivalent to ω( k1 ), we can write
Z
Z
u
θ
u
θ
∇u∇ϕ1 H 2 ( )H 2 ( )
−
uϕ1 H 2 ( )H 2 ( ) − D1
k
k
k
k
Ω
ZΩ
θ
u
1
− [a|∇u|2 + b|∇v|α ]ϕ1 H 2 ( )H 2 ( ) + ω( )
k
k
k
Ω
Z
θ
u
≤−
f ϕ1 H 2 ( )H 2 ( )
k
k
Ω
Finally passing to the limit as k tends to infinity, we use the fact that
θ
lim H( ) = 1,
k
u
lim H( ) = 1
k
k→∞
k→∞
to conclude that for every ϕ1 ≥ 0, ϕ1 ∈ D(Ω),
Z
Z
2
α
[u − D1 ∆u + a(x)|∇u| + b(x)|∇v| ]ϕ1 ≥
f ϕ1
Ω
Ω
In the same way, we multiply the second equation in (3.2) by Φ2 and we integrate
on Ω. By studying separately each term as in the previous case still using Lemmas
3.1, 3.3 and 3.4, we choose
ψ2 = −ϕ2 exp[
θ
θ
v
u]H( )H( )
D2
k
k
where ϕ2 ≥ 0, ϕ2 ∈ D(Ω) and we replace ψ2 by this value in the inequality obtained
to conclude that for every ϕ2 ≥ 0, ϕ2 ∈ D(Ω) that
Z
Z
β
2
[v − D2 ∆v + c(x)|∇u| + d(x)|∇v| ]ϕ2 ≥
gϕ2
Ω
Ω
This completes the proof of theorem 2.2.
Acknowledgments. We are grateful to the anonymous referee for the corrections
and useful suggestions that have improved this article.
12
A. MOUIDA, N. ALAA, S. MESBAHI, W. BOUARIFI
EJDE-2013/142
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Abdelhaq Mouida
Laboratory LAMAI, Faculty of Science and Technology of Marrakech, University Cadi
Ayyad, B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco
E-mail address: abdhaq.mouida@gmail.com
Noureddine Alaa
Laboratory LAMAI, Faculty of Science and Technology of Marrakech, University Cadi
Ayyad, B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco
E-mail address: n.alaa@uca.ma
Salim Mesbahi
Department of Mathematics, Faculty of Science, University Ferhat Abbas, Setif I, Pole
2 - Elbez, Setif - 19000, Algeria
E-mail address: salimbra@gmail.com
Walid Bouarifi
Department of Computer Science, National School of Applied Sciences of Safi, Cadi
Ayyad University, Sidi Bouzid, BP 63, Safi - 46000, Morocco
E-mail address: w.bouarifi@uca.ma