Electronic Journal of Differential Equations, Vol. 2001(2001), No. 19, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Local and global estimates for solutions of
systems involving the p-Laplacian
in unbounded domains ∗
A. Bechah
Abstract
In this paper, we study the local and global behavior of solutions of
systems involving the p-Laplacian operator in unbounded domains. We
extend some Serrin-type estimates which are known for simple equations
to systems of equations.
1
Introduction
We consider the system
−∆p u = f (x, u, v) x ∈ Ω,
−∆q v = g(x, u, v) x ∈ Ω,
u = v = 0 x ∈ ∂Ω.
(1.1)
(1.2)
(1.3)
where Ω ⊂ RN is an exterior domain, f, g are a given functions depending of
the variables x, u, v and ∆p is the p-Laplacian
operator; for 1 < p < +∞ ∆p
is defined by ∆p u = div |∇u|p−2 ∇u . Here, we study the local and global
behavior of solutions of System (1.1)–(1.3). we follow the work of Serrin [4]
concerning the quasilinear equation
div A(x, u, ux ) = B(x, u, ux ),
(1.4)
where A and B are a given functions depending of the variables x, u, ux and
∂u
∂u
, . . . , ∂x
). In particular, (1.4) generalizes the equation
ux = ( ∂x
1
n
−∆p u = f (x, u) x ∈ Ω.
(1.5)
p−1
In [4], Serrin proves that if the function f is bounded by the term a|u|
+ g,
where p > 1 is a fixed exponent, a is a positive constant and g is a measurable
function, then for each y ∈ Ω and R > 0 we have the estimate
N
1
N
N
sup u(x) ≤ cR− p kukLp (B2R (y)) + R p (R kgk p−
) p−1
(1.6)
L
BR (y)
∗ Mathematics
(B2R (y))
Subject Classifications: 35J20, 35J45, 35J50, 35J70.
Key words: quasilinear systems, p-Laplacian operator, unbounded domain, Serrin estimate.
c
2001
Southwest Texas State University.
Submitted November 23, 2001. Published March 23, 2001.
1
2
Local and global estimates for solutions
EJDE–2001/19
for all 0 < ≤ 1.
In many cases, especially for unbounded domain, when we wish to show
that the solution decay at infinity, the estimate (1.6) requires that the function
f belongs to Lα (Ω) with α > N/p, which is not trivial to prove in some cases. To
avoid this difficulty Yu [5], Egnell [1] and others have proved that the solution
of (1.5) have a regularity Lq (Ω) for each q ≥ p∗ , and this for all function f
bounded by a sublinear, superlinear or an homogeneous terms. We note that in
the case of a mixed terms this last technique cannot be adapted. For the case of
an homogeneous system see the paper of Fleckinger, Manàsevich, Stavrakakis
and de Thélin [2].
The first part of this paper is devoted to the local behavior of solutions of
System (1.1)–(1.3). We obtain an estimate of Serrin type in the following cases:
1) f and g are bounded by a sum of homogeneous and critical terms.
2) f and g are bounded by a sum of homogeneous and constant terms.
Thus, we extend the results of [5], [1] concerning Equation and those of [2]
concerning System.
In the second part, we obtain a global estimates of solutions of System (1.1)–
(1.3) in the particular case f = A|u|α−1 u|v|β+1 and g = B|u|α+1 |v|β−1 v under
some conditions on α, β, p and q. Also we obtain another global estimate when
f and g satisfy 2).
We recall that D1,p (Ω) is the closure of C0∞ (Ω) with respect to the norm
kukD1,p (Ω) = k∇ukLp (Ω) .
p
p
p0 = p−1
is the conjugate of p, p∗ = NN−p
is the Sobolev exponent and we define
Sp by
(
)
k∇ukpLp (Ω)
1
1,p
= inf
u ∈ W (Ω)\{0} .
Sp
kukpLp (Ω)
2
Local estimates for solutions of (1.1)–(1.3)
Theorem 2.1 Let (u, v) ∈ D1,p (RN ) × D1,q (RN ) be a solution of (1.1) − (1.3)
and τ = NN−p , τ̄ = NN−q . Assume that max{p, q} < N , q ≥ p and
and
τq
∗
0
|f (x, u, v)| ≤ C |u|p−1 + |u|p −1 + |v|q/p + |v| (τ p)0 ,
(2.1)
τp
0
|g(x, u, v)| ≤ C |v|q−1 + |v|τ q−1 + |u|p/q + |u| (τ q)0 ,
(2.2)
where m0 is the conjugate of m and C is a constant. Then
1) For any R > 0 and x ∈ RN satisfying
o
n
q−p
C max 2p Sp τ p−1 , 22q−p Sq |B1 | N Rq−p τ q−1
p(τ −1)
q(τ −1)
× kukLp∗ (B2R (x)) + kvkLqτ (B2R (x)) < 1
(2.3)
EJDE–2001/19
A. Bechah
3
where Sp and Sq are the Sobolev constants, we have
kukL∞ (B R (x))
2
≤ c (1 + Rq )
N (N −p)
p3
n p−N
o
q−N
q/p
max R p kukLp∗ (BR (x)) , R p kvkLq∗ (BR (x)) .
N (N −p)
qp2
n q−N
o
p
p−N
max R q kvkLq∗ (BR (x)) , R q kukLq p∗ (BR (x)) .
and
kvkL∞ (B R (x) )
2
≤ c (1 + Rq )
witch c independent of u, v, x and R.
2) Moreover,
lim u(x) =
|x|→+∞
lim
|x|→+∞
v(x) = 0.
Remark 2.2 There exists an R0 such that for all R < R0 , (2.3) is satisfied
uniformly forR all x ∈ Ω. This follows
from the absolute continuity of the funcR
∗
tionals A 7→ A |u|p dx and A 7→ A |v|qτ dx. To be more specific, for each > 0
there exists
that forRall R > 0 and x ∈ RN satisfying |BR (x)| ≤ η,
R η > 0 such
∗
we have BR (x) |u|p dx < and BR (x) |v|qτ dx < .
Proof Let x ∈ RN be fixed. For y ∈ B2R (x) and any function h defined on
B2R (x) we define
y−x
.
h̃(t) = h(y), t =
R
Since (u, v) is a solution for (1.1)–(1.3), then (ũ, ṽ) satisfies
−∆p ũ = Rp f (y, ũ, ṽ),
(2.4)
−∆q ṽ = Rq g(y, ũ, ṽ).
(2.5)
In this proof c denotes a positive constant independent of u, v, x and R. For
any ball B ⊂ B2 (0), we have
∀w ∈ W01,p (B) kwkpLpτ (B) ≤ Sp k∇wkpLp (B) ,
∀w ∈ W01,q (B) kwkqLqτ (B) ≤ 2q−p |B1 (0)|
q−p
N
Sq k∇wkqLq (B) .
(2.6)
Sp and Sq are the Sobelev constants. Let (mn )n be a sequence of positive
numbers satisfying σ < ∞ where σ is defined below and (rn )n a decreasing
sequence defined by
0
r0 = 2,
−1/p
n−1 1 X mi + p
rn = 2 −
,
σ i=0
p
4
Local and global estimates for solutions
where R is positive and σ =
−1/p
∞ X
mi + p
p
i=0
EJDE–2001/19
0
. We denote by Bn = B(0, rn )
and we define η ∈ C0∞ (RN ) so that 0 ≤ η ≤ 1, η = 1 in Bn+1 , supp(η) ⊂ Bn
and
1/p0
mn + p
|∇η| ≤ c
.
(2.7)
p
We multiply (2.4) by |ũ|mn ũη q , and integrate over Bn . Using (2.1), we obtain
I1 + I2 ≤ Rp (I3 + I4 + I5 + I6 ) ,
(2.8)
where
Z
I1 = (1 + mn )
η q |ũ|mn |∇ũ|p dx,
Bn
Z
I2 = q
η q−1 ∇η.∇ũ|∇ũ|p−2 |ũ|mn ũdx,
Bn
Z
I3 = C
|ũ|p+mn η q dx,
Bn
Z
∗
I4 = C
|ũ|p +mn η q dx,
ZBn
0
I5 = C
|ũ|mn ũ|ṽ|q/p η q dx,
ZBn
τq
I6 = C
|ũ|mn ũ|ṽ| (τ p)0 η q dx.
Bn
n
Since 1 + mn = (p−1)m
+ mnp+p , we deduce from Young inequality and the facts
p
p ≤ q, |η| ≤ 1, that for any s > 0
0
|I2 |
≤
qsp
p0
Z
mn + p
η q |∇ũ|p |ũ|mn dx
p
Bn
− p0 Z
p
q
mn + p
+ p
|∇η|p |ũ|mn +p dx
ps
p
Bn
0
Choosing s such that
qsp
p0
≤ 12 , and using (2.7), we have
|I2 | ≤
1
I1 + c
2
Z
|ũ|mn +p dx.
(2.9)
Bn
We deduce from (2.8) and (2.9)
I1 ≤ 2R
p
6
X
i=3
Ii + c
Z
Bn
|ũ|mn +p dx.
(2.10)
EJDE–2001/19
A. Bechah
5
Using Sobolev inequality and observing that for any a ≥ 0 and b ≥ 0 (a + b)p ≤
2p−1 (ap + bp ), we have
q/p mnp+p p
≤ 2p−1 Sp (I7 + I8 ) ,
(2.11)
η ũ
pτ
L
(Bn )
where
q
I7 = ( )p
p
Z
η q−p |∇η|p |ũ|mn +p dx ≤ c
Bn
and
I8 =
mn + p
p
p Z
mn + p
p
η q |ũ|mn |∇ũ|p dx ≤
Bn
p−1 Z
|ũ|mn +p dx,
Bn
mn + p
p
p−1
I1 ,
thus we deduce from (2.10) that
q/p mnp+p p
η ũ
pτ
L
(Bn )
≤
mn + p
p
p−1
c
Z
mn +p
|ũ|
p
dx + 2 Sp R
p
Bn
6
X
Ii
!
.
i=3
(2.12)
First step. We construct the sequences (pn )n and (qn )n by
pn = pτ n ,
qn = qτ n ,
and we set
mn = p(τ n − 1), and
ln = q(τ n − 1).
We show that if the condition
n
o
q−p
C max 2p Sp Rp τ n(p−1) , 22q−p |B1 | N Sq Rq τ n(q−1)
p(τ −1)
q(τ −1)
× kũkLp∗ (B2 ) + kṽkLqτ (B2 )
< 1,
is satisfied, the solution (ũ, ṽ) belongs to Lpn+1 (Bn+1 ) × Lqn+1 (Bn+1 ).
First, we start by estimating the integrals (Ii ), i = 3, . . . , 6. We have
Z
I3 = C
|ũ|p+mn η q dx ≤ ckũkpLnpn (Bn ) .
(2.13)
Bn
Remarking that
mn +1
pn
I5 = C
Z
Bn
+
q
p0
qn
= 1, we deduce from Hölder inequality that
0
q/p0
n +1
|ũ|mn ũ|ṽ|q/p η q dx ≤ ckũkm
Lpn (Bn ) kṽkLqn (Bn ) .
(2.14)
n +p
We write mn + p∗ = p(τ − 1) + mn + p , q = τ q( mpn+1
). Observing that
mn +p
pn+1
+
p(τ −1)
p∗
= 1, we deduce from Hölder inequality
Z
Z
∗
τ q( mn +p )
I4 =C
|ũ|p +mn η q dx ≤ C
|ũ|p(τ −1) |ũ|p+mn η pn+1 dx
Bn
Bn
p(τ −1)
q/p τ n p
≤CkũkLp∗ (Bn ) kη ũ kLpτ (Bn ) .
(2.15)
6
Local and global estimates for solutions
EJDE–2001/19
Remark that
q
q(τ − 1) mn + 1
p0
+
+
= 1,
τq
pn+1
qn+1
τq
q
− 0 = q(τ − 1),
(τ p)0
p
and
(2.16)
q
mn + 1
p0
+τ
= 1,
pn+1
qn+1
then from Hölder inequality, we have
Z
τq
I6 =C
|ũ|mn ũ|ṽ| (τ p)0 η q dx
τ
Bn
≤C
Z
mn +1
q(τ −1) τ q( pn+1 )
|ṽ|
η
q
0
p
mn +1 τ q( qn+1 )
|ũ|
η
0
|ṽ|q/p dx
(2.17)
Bn
q
0
1+mn
p
n p(
n q(
)
q(τ −1)
pn )
≤CkṽkLτ q (Bn ) kη q/p ũτ kLpτ (B
kηṽ τ kLqτqn(Bn ) .
n)
Substituting mn by p(τ n − 1) in (2.12), we obtain
n
kη q/p ũτ kpLpτ (Bn ) − τ n(p−1) 2p Sp Rp (I4 + I6 )
Z
n(p−1)
pn
p
p
≤τ
c
|ũ| dx + 2 Sp R (I3 + I5 ) .
(2.18)
Bn
It follows from (2.13) - (2.17) and the fact p ≤ q that
n
n
p(τ −1)
kη q/p ũτ kpLpτ (Bn ) − C2p Sp Rp τ n(p−1) kũkLp∗ (Bn ) kη q/p ũτ kpLpτ (Bn )
!
q
q(τ −1)
n
p
(1+mn )
n
q(
p0
)
n
+kṽkLτ q (Bn ) kη q/p ũτ kLpτ p(B
kηṽ τ kLqτqn(Bn )
n)
(2.19)
q/p0
n +1
≤ c(1 + Rq )τ n(q−1) kũkpLnpn (Bn ) + kũkm
kṽk
p
q
L n (Bn )
L n (Bn ) .
Similarly, we have
n
kηṽ τ kqLqτ (Bn ) − C22q−p Sq |B1 |
q−p
N
n
q(τ −1)
Rq τ n(q−1) kṽkLqτ (Bn ) kηṽ τ kqLqτ (Bn )
p
0
(1+l )
+
q
n
n q
n p(
)
p(τ −1)
n
kũkLτ p (Bn ) kηṽ τ kLqτq(B
kη q/p ũτ kLpτpn(Bn )
n)
(2.20)
p/q 0
≤ c(1 + Rq )τ n(q−1) kṽkqLnqn (Bn ) + cRq τ n(q−1) kṽklLnq+1
n (Bn ) kũkLpn (Bn ) .
n
n
Next, we define θn+1 = max{kη q/p ũτ kpLpτ (Bn ) , kηṽ τ kqLqτ (Bn ) }, and
En = max{kũkpLnpn (Bn ) , kṽkqLnqn (Bn ) }1/pn . Simple computations using Hölder
inequality and the definition of En and θn , show that
n
o
q−p
θn+1 − C max 2p Sp Rp τ n(p−1) , 22q−p |B1 | N Sq Rq τ n(q−1)
(2.21)
p(τ −1)
q(τ −1)
× kũkLp∗ (Bn ) + kṽkLqτ (Bn ) θn+1 ≤ c(1 + Rq )τ n(q−1) Enpn .
EJDE–2001/19
A. Bechah
7
We know that there exists R0 > 0 such that for any R < R0
n
o
q−p
C max 2p Sp Rp τ n(p−1) , 22q−p |B2 | N Sq Rq τ n(q−1)
p(τ −1)
q(τ −1)
× kũkLp∗ (B2 ) + kṽkLqτ (B2 ) < 1.
(2.22)
Also, remark that
θn+1 ≥ max{kũkpLnpn+1 (Bn+1 ) , kṽkqLnqn+1 (Bn+1 ) }
p
q
≥ max{kũkLn+1
, kṽkLn+1
}1/τ
pn+1
qn+1
(Bn+1 )
(Bn+1 )
=
(2.23)
pn
En+1
.
Therefore, from (2.21) - (2.23), and the fact p ≤ q
pn
En+1
≤ c(1 + Rq )τ n(q−1) Enpn .
So
1/pn
En+1 ≤ (c(1 + Rq ))
τ
n(q−1)
pn
En .
This implies that
P∞
kũkLpn+1 (Bn+1 ) ≤ En+1 ≤ (c(1 + Rq ))
P∞
P∞
Since i=0 pτ1 i = pN2 and i=0
Similarly, we have
q/p
i(q−1)
pτ i
1
i=0 pτ i
τ
P∞
i=0
i(q−1)
pτ i
E0 .
< ∞, we deduce that ũ ∈ Lpn+1 (Bn+1 ).
qn+1
p
P∞
kṽkLqn+1 (Bn+1 ) ≤ kṽkLn+1
≤ En+1 ≤ (c(1 + Rq ))
qn+1
(Bn+1 )
1
i=0 pτ i
τ
P∞
i=0
i(q−1)
pτ i
E0 ,
therefore v ∈ Lqn+1 (Bn+1 ).
Second step We remark that hypothesis (2.3) is equivalent to
n
o
q−p
p(τ −1)
q(τ −1)
C max 2p Sp Rp τ p−1 , 22q−p |B1 | N Sq Rq τ q−1 kũkLp∗ (B2 ) + kṽkLq∗ (B2 ) < 1.
We assume that R, u and v satisfy (2.3), which by the first step implies that
2
2
τ
τ2
(ũ, ṽ) ∈ Lpτ (B1 ) × Lqτ (B1 ). We let δ = τ 2 −τ
+1 and χ = δ . It is clear that
1 < δ < τ , and so χ > 1. We construct a sequences (sn )n and (tn )n by
sn = pχn ,
tn = qχn .
In this step mn and rn are defined by
n
χ
mn = p
−1 ,
δ
and
0
r0 = 1,
−1/p
n−1 1 X mi + p
rn = 1 −
,
2σ i=0
p
8
Local and global estimates for solutions
EJDE–2001/19
which implies mn + p = sn /δ. Now, we estimate the integrals (Ii )i=3,...,6 . We
have
s /δ
s /δ
I3 ≤ ckũk nsn
≤ ckũkLnsn (Bn ) .
(2.24)
L
Remarking that
mn +1
sn /δ
q/p0
tn /δ
+
I5 ≤ ckũkmsnn+1
L
We have
p(τ −1)
pτ 2
+
mn +p
sn
δ
(Bn )
δ
= 1, it follows from Hölder inequality that
(Bn )
kṽk
q/p0
q/p0
tn
L δ
(Bn )
n +1
≤ kũkm
Lsn (Bn ) kṽkLsn (Bn ) .
(2.25)
= 1, thus from Hölder inequality we have
p(τ −1)
s /δ
s /δ
I4 ≤ ckũkLpτ 2 (B ) kũkLnsn (Bn ) ≤ ckũkLnsn (Bn ) .
(2.26)
n
Observing that
q(τ −1)
qτ 2
0
+ msnn+1 + q/p
tn = 1, it follows from Hölder inequality that
I6 ≤ c
Z
0
|ṽ|q(τ −1) |ũ|mn +1 |ṽ|q/p dx
Bn
q/p0
q(τ −1)
n +1
≤ ckṽkLqτ 2 (B ) kũkm
Lsn (Bn ) kṽkLtn (Bn )
(2.27)
n
q/p0
n +1
≤ ckũkm
Lsn (Bn ) kṽkLtn (Bn ) .
We deduce from (2.12), (2.24)–(2.27) and the fact p ≤ q that
q/p χn /δ p
s /δ
q/p0
n +1
≤ cχn(q−1) (1 + Rq ) kũkLnsn (Bn ) + kũkm
η ũ
pτ
Lsn (Bn ) kṽkLtn (Bn )
L
(Bn )
(2.28)
Similarly, we have
χn /δ q
t /δ
p/q 0
≤ cχn(q−1) (1 + Rq ) kṽkLntn (Bn ) + kṽklLnt+1
kũk
ηṽ
qτ
s
n (Bn )
L n (Bn )
L
(Bn )
(2.29)
n
o1/sn
As in the first step, we let Λn = max kũksLnsn (Bn ) , kṽktLntn (Bn )
n
o
n
n
Γn = max kη q/p ũχ /δ kpLpτ (Bn ) , kηṽ χ /δ kqLqτ (Bn ) and
n
o t1
n
. Simple computations show that
Υn = max kũksLnsn (Bn ) , kṽktLntn (Bn )
and
n
o
q/p0
sn /δ
tn /δ
n +1
kũkm
kṽk
≤
min
Λ
,
Υ
,
s
t
n
n
L n (Bn )
L n (Bn )
(2.30)
n
o
p/q 0
sn /δ
tn /δ
kṽklLnt+1
kũk
≤
min
Λ
,
Υ
.
n (Bn )
n
n
Lsn (Bn )
(2.31)
Also, remark that
n
s /δ
t /δ
Γn ≥ max kũkLnsn+1 (Bn+1 ) , kṽkLntn+1 (B
n+1 )
o
s /δ
n
= Λn+1
= Υtnn /δ .
(2.32)
EJDE–2001/19
A. Bechah
9
Thus, we deduce from (2.28)–(2.32) that
s /δ
n
Λn+1
≤ cχn(q−1) (1 + Rq ) Λsnn /δ ,
and so
Λn+1 ≤ cδ/sn χ
n(q−1)δ
sn
δ/sn
(1 + Rq )
Λn .
Which implies that
P∞
kũkLsn (Bn ) ≤ Λn ≤ c
Since
P∞
δ
i=0 si
δτ
p(τ −δ) ,
=
and
δ
i=0 si
P∞
i=0
P∞
χ
i=0
i(q−1)δ
si
i(q−1)δ
si
P∞
(1 + Rq )
δ
i=0 si
Λ0 .
< ∞, then
kũkL∞ (B 1 ) ≤ lim sup kũkLsn (Bn )
n→+∞
2
n
o
δτ
q/p
≤ c (1 + Rq ) p(τ −δ) max kũkLp (B1 ) , kṽkLq (B1 ) .
Similarly, we have
δ
Υn+1 ≤ c tn χ
n(q−1)δ
tn
δ
(1 + Rq ) tn Υn
As n tends to infinity, we obtain
kṽkL∞ (B 1 ) ≤ lim sup kṽkLtn (Bn )
n→+∞
2
n
o
p
δτ
≤ c (1 + Rq ) q(τ −δ) max kṽkLp (B1 ) , kũkLq q (B1 ) .
By the imbeddings
∗
Lp (B1 ) ⊂ Lp (B1 )
and the fact
∗
and Lq (B1 ) ⊂ Lq (B1 ),
δτ
τ
N (N − p)
=
=
,
2
τ −δ
(τ − 1)
p2
we have
kũkL∞ (B 1 ) ≤ c (1 + Rq )
N (N −p)
p3
n
o
q/p
max kũkLp∗ (B1 ) , kṽkLq∗ (B1 ) ,
N (N −p)
qp2
n
o
p
max kṽkLp∗ (B1 ) , kũkLq q∗ (B1 ) .
2
and
kṽkL∞ (B 1 ) ≤ c (1 + Rq )
2
Coming back to (u, v) by a simple change of variables, we find
kukL∞ (B R (x) )
2
≤ c (1 + Rq )
N (N −p)
p3
n p−N
o
q−N
q/p
max R p kukLp∗ (BR (x)) , R p kvkLq∗ (BR (x)) .
10
Local and global estimates for solutions
EJDE–2001/19
and
kvkL∞ (B R (x) )
2
≤ c (1 + Rq )
N (N −p)
qp2
q−N
o
p
p−N
max R q kvkLq∗ (BR (x)) , R q kukLq p∗ (BR (x)) .
The proof of 2) follows from 1) and Remark 2.2
Proposition 2.3 Let (u, v) ∈ D
We assume q ≥ p,
and
1,p
N
(R ) × D
1,q
♦
N
(R ) a solution of (1.1)–(1.3).
0
|f (x, u, v)| ≤ C |u|p−1 + |v|q/p + 1 ,
(2.33)
0
|g(x, u, v)| ≤ C |v|q−1 + |u|p/q + 1 ,
(2.34)
0
where m is the conjugate of m. Then
n
o
N
p−N
q−N
q/p
kukL∞ (B1 ) ≤ c (1 + Rq ) p2 max 1, R p kukLp∗ (B2 ) , R p kvkLq∗ (B2 ) , (2.35)
and
n
o
p
N
q−N
p−N
kvkL∞ (B1 ) ≤ c (1 + Rq ) pq max 1, R q kukLq p∗ (B2 ) , R q kvkLq∗ (B2 ) . (2.36)
Proof We use the same change of variables as in the proof of Theorem 2.1.
Thus, we obtain that (ũ, ṽ) satisfies (2.4) and (2.5). Also we keep the same
sequences (mn )n , (rn )n , (Bn )n and the same function η. We multiply Equation
(2.4) by |ũ|mn ũη q , and integrate over Bn . Using (2.33), we have
I1 + I2 ≤ Rp (I3 + I4 + I5 ) ,
(2.37)
where
Z
I1 = (1 + mn )
η q |ũ|mn |∇ũ|p dx,
Bn
Z
I2 = q
η q−1 ∇η.∇ũ|∇ũ|p−2 |ũ|mn ũdx,
Bn
Z
I3 = C
|ũ|p+mn η q dx,
Bn
Z
0
I4 = C
|ũ|mn ũ|ṽ|q/p η q dx,
ZBn
I5 = C
|ũ|mn ũη q dx.
Bn
The integrals I1 , I2 , I3 and I4 are the same to those obtained in Theorem 2.1.
Simple computations used before show that
!
p−1
Z
5
X
mn + p
q/p mnp+p p
mn +p
p
p
≤
c
|ũ|
dx + 2 Sp R
Ii .
η ũ
pτ
p
L (Bn )
Bn
i=3
(2.38)
EJDE–2001/19
A. Bechah
11
Now, we define (pn )n and (qn )n by
pn = pτ n ,
qn = qτ n ,
and let mn = p(τ n − 1), and , ln = q(τ n − 1). Then we estimate the integrals
Ii , i = 3, . . . , 5. It is clear from (2.13) and (2.14) that
I3 ≤ ckũkpLnpn (Bn )
q/p0
n +1
and I4 ≤ ckũkm
Lpn (Bn ) kṽkLqn (Bn ) .
(2.39)
On the other hand
Z
( m1n − p1n )(mn +1)
mn +1
n +1
I5 ≤ C
|ũ|mn +1 dx = ckũkL
kũkm
mn +1 (B ) ≤ c|Bn |
Lpn (Bn )
n
Bn
p−1
n +1
≤ c|B2 | pτ n kũkm
Lpn (Bn )
n +1
≤ ckũkm
Lpn (Bn ) .
(2.40)
We deduce from (2.38)–(2.40) that
n
kũkpLnpn+1 (Bn+1 ) ≤ kη q/p ũτ kpLpτ (Bn )
≤ cτ n(p−1) kũkpLnpn (Bn )
q/p0
mn +1
mn +1
+ Rp kũkpLnpn (Bn ) + kũkL
.
pn (B ) kṽkLqn (B ) + kũkLpn (B )
n
n
n
(2.41)
Similarly, we have
n
kṽkqLnqn+1 (Bn+1 ) ≤ kηṽ τ kqLqτ (Bn )
≤ cτ n(q−1) kṽkqLnqn (Bn )
p/q 0
ln +1
+ Rq kṽkqLnqn (Bn ) + kṽklLnq+1
.
n (Bn ) kũkLpn (Bn ) + kṽkLqn (Bn )
(2.42)
Following the proof of Theorem 2.1 we let
n
o1/pn
En = max 1, kũn kpLnpn (Bn ) , kṽn kqLnqn (Bn )
and
n
o q1
n
Fn = 1, kũn kpLnpn (Bn ) , kṽn kqLnqn (Bn )
. We obtain
kũkL∞ (B1 ) ≤ lim sup kũkLpn (Bn ) ≤ En
n→+∞
N
≤ c (1 + Rq ) p2 E0
= c (1 + Rq )
N
p2
n
o
q/p
max 1, kũkLp (B2 ) , kṽkLq (B2 ) .
(2.43)
12
Local and global estimates for solutions
EJDE–2001/19
kṽkL∞ (B1 ) ≤ lim sup kṽkLqn (Bn ) ≤ Fn
n→+∞
N
≤ c (1 + Rq ) pq F0
= c (1 + Rq )
N
pq
(2.44)
n
o
p
max 1, kũkLq p (B2 ) , kṽkLq (B2 ) .
Using a simple change of variables in (2.43) and (2.44) we obtain (2.35) and
(2.36).
3
Global estimates for solutions of (1.1)–(1.3)
Proposition 3.1 Let (u, v) ∈ D1,p (Ω) × D1,q (Ω) a solution of (1.1)–(1.3). We
assume that there exist a functions a, b ∈ L1 (Ω) ∩ L∞ (Ω) and a constant C such
that
0
|f (x, u, v)| ≤ a(x) + C(|u|p−1 + |v|q/p ),
(3.1)
0
|g(x, u, v)| ≤ b(x) + C(|v|q−1 + |v|p/q ),
(3.2)
where p > 1, q > 1. Then
1) (u, v) ∈ Lσ (Ω) × Lη (Ω) for all (σ, η) ∈ [p∗ , +∞) × [q ∗ , +∞).
2) lim u(x) = lim v(x) = 0.
|x|→+∞
|x|→+∞
Proof 1) Let pn = pτ n , qn = qτ n , mn = τ n − 1, tn = τ n − 1, Tk (u) =
max{−k, min{k, u}} and w = |Tk (u)|pmn Tk (u), with k > 0. Multiplying the
equation (1.1) by w and integrating over Ω, we obtain
Z
Z
p
pmn
(pmn + 1)
|∇Tk (u)| |Tk (u)|
dx =
f (x, u, v)w dx.
Ω
Ω
Observing that
(
1
)p |∇(Tk (u))mn +1 |p = Tk (u)pmn |∇Tk (u)|p ,
mn + 1
(3.3)
we deduce from Hölder and Sobolev inequalities that for any 0 < γ < 1, we
have
Z
|Tk (u)|τ (pmn +p) )1/τ
Ω
(3.4)
q/p0
γ
pmn +1
pn
mn +1
≤ c kak1−γ
∞ kakL1 (Ω) kukLpn (Ω) + kukLpn (Ω) + kvkLqn (Ω) kukLpn (Ω) .
with c depending from n. Letting k tend to infinity in (3.4), we obtain
q/p0
pn
mn +1
n +1
kukpLnpn+1 (Ω) ≤ c kukpm
Lpn (Ω) + kukLpn (Ω) + kvkLqn (Ω) kukLpn (Ω) .
(3.5)
We derive from (3.5) that u ∈ Lpn (Ω) for all n ∈ N. Similarly, we prove that
v ∈ Lqn (Ω) for all n ∈ N. By interpolation inequality (see [3]) we prove that
EJDE–2001/19
A. Bechah
13
(u, v) ∈ Lσ (Ω) × Lη (Ω), for all (σ, η) ∈ [p∗ , +∞) × [q ∗ , +∞). The proof of 2)
follows from Serrin inequality [4] and 1).
♦
Next, we study the sub-homogeneous system
−∆p u = B(x)|u|α−1 u|v|β+1 ,
α+1
−∆q v = C(x)|u|
β−1
|v|
v,
(3.6)
(3.7)
in Ω an exterior domain or RN .
Proposition 3.2 Assume that B, C ∈ L∞ (Ω) and
α+1 β+1
+
< 1,
p∗
q∗
p > 1,
q > 1.
Then each solution (u, v) ∈ D1,p (Ω)×D1,q (Ω) of the system (3.6), (3.7) satisfies
1. (u, v) ∈ Lσ (Ω) × Lη (Ω) for all (σ, η) ∈ [p∗ , +∞[×[q ∗ , +∞[.
2. lim|x|→+∞ u(x) = 0 and lim|x|→+∞ v(x) = 0.
β+1
Proof Let τ = NN−p , τ̄ = NN−q and L = 1 − α+1
p∗ − q ∗ . Assume q ≥ p, which
implies that τ̄ ≥ τ . We define the sequences (pn )n , (qn )n and (fn )n by
fn+1 = τ (fn + L − 1) + 1,
pn = p∗ fn , qn = q ∗ fn .
f0 = 1,
Let Tk (u) = max{−k, min{k, u}} for k > 0 and ω = |Tk (u)|pmn Tk (u), with
mn = (1 −
α + 1 β + 1 pn
−
)
= fn+1 − 1
pn
qn
p
(3.8)
Multiplying (3.6) by ω and integrating over Ω, we obtain from (3.3) and Sobolev
inequality
Z
Z
1
1
(pmn + 1)(
)p ( |Tk (u)|τ (pmn +p) )1/τ ≤ kBkL∞ (Ω)
|u|α |v|β+1 ωdx.
Sp
mn + 1
Ω
Ω
From the definition of mn and Hölder inequality, we deduce that
Z
∗
(mn + 1)p
n
( |Tk (u)|p (mn +1) )1/τ ≤ Sp
kBkL∞ (Ω) kukα+1+pm
kvkβ+1
Lpn (Ω)
Lqn (Ω) .
(pmn + 1)
Ω
Let k tends to infinity, we have
Z
∗
(mn + 1)p
n
( |u|p (mn +1) )1/τ ≤ Sp
kBkL∞ (Ω) kukα+1+pm
kvkβ+1
Lpn (Ω)
Lqn (Ω) .
(pmn + 1)
Ω
p∗ (mn + 1) = p∗ (fn+1 ) = pn+1 , therefore u ∈ Lpn+1 (Ω). To show that v ∈
Lqn+1 (Ω), We consider w̄ = |Tk (v)|qtn Tk (v), with
α + 1 β + 1 qn
−
)
pn
qn
q
= τ̄ (fn + L − 1).
tn = (1 −
(3.9)
14
Local and global estimates for solutions
EJDE–2001/19
Proceeding as above, we obtain
Z
∗
1
(tn + 1)q
β+1+qtn
( |v|q (tn +1) ) τ̄ ≤ Sq
kCkL∞ (Ω) kukα+1
Lpn (Ω) kvkLqn (Ω) .
(qtn + 1)
Ω
Let q̄n = q ∗ (tn + 1). It is clear that v ∈ Lq̄n , and since
q̄n = q ∗ (tn + 1)
= q ∗ (τ̄ (fn + L − 1) + 1)
≥ qn+1 ,
then qn ≤ qn+1 ≤ q̄n . By interpolation inequality (see [3]), we deduce that
v ∈ Lqn+1 (Ω). 2) follows from Serrin inequality [4] and 1).
References
[1] E. Egnell. Existence results for some quasilinear elliptic equations. Nonlinear. Anal, 1989.
[2] J. Fleckinger, R.F. Manasevich, N.M. Stavrakakis, and F. de Thelin. Principal eigenvalues for some quasilinear elliptic equations on Rn . Adv.
Differential Equations, 2(6):981–1003, 1996.
[3] D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations
of Second Order. Springer-Verlag, 1983.
[4] J. Serrin. Local behavior of solutions of quasilinear equations. Acta. Math,
111:247–302, 1964.
[5] L.S. Yu. Nonlinear p-Laplacian problem on unbounded domains. Proc.
Amer. Math. Soc, 115(4):1037–1045, 1992.
Abdelhak Bechah
Laoratoire MIP
UFR MIG, Université Paul Sabatier Toulouse 3
118 route de Narbonne, 31 062
Toulouse cedex 4, France
e-mail: bechah@mip.ups-tlse.fr or abechah@free.fr
web page: http://mip.ups-tlse.fr/∼bechah