2 Local estimates for solutions of (1.1)–(1.3)

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

−∆pu=f(x, u, v) x∈Ω, (1.1)

−∆qv=g(x, u, v) x∈Ω, (1.2)

u=v= 0 x∈∂Ω. (1.3)

where Ω ⊂R^N 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 ∆pu = 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

divA(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 ux= (_∂x^∂u

1, . . . ,_∂x^∂u

n). In particular, (1.4) generalizes the equation

−∆pu=f(x, u) x∈Ω. (1.5)

In [4], Serrin proves that if the function f is bounded by the terma|u|^p−1+g, where p >1 is a fixed exponent, ais a positive constant andg is a measurable function, then for each y∈Ω andR >0 we have the estimate

sup

B_R(y)

u(x)≤cR^−Np

kukL^p(B2R(y))+R^Np(Rkgk

L^p−N (B_2R(y)))^p−11

(1.6)

∗Mathematics Subject Classifications: 35J20, 35J45, 35J50, 35J70.

Key words:quasilinear systems, p-Laplacian operator, unbounded domain, Serrin estimate.

2001 Southwest Texas State University.c

Submitted November 23, 2001. Published March 23, 2001.

1

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 toL^α(Ω) withα > 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 L^q(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, Man`asevich, Stavrakakis and de Th´elin [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 andg are bounded by a sum of homogeneous and critical terms.

2) f andg 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 casef =A|u|^α−1u|v|^β+1 and g=B|u|^α+1|v|^β−1v under some conditions onα, β, pandq. Also we obtain another global estimate when f andg satisfy 2).

We recall thatD^1,p(Ω) is the closure ofC0^∞(Ω) with respect to the norm kuk_D^1,p(Ω)=k∇ukL^p(Ω).

p⁰=_p^p

−1 is the conjugate ofp,p∗= _N^{N p}

−p is the Sobolev exponent and we define S_p by

1 Sp

= inf

(k∇uk^p_Lp(Ω)

kuk^p_Lp(Ω)

u∈W^1,p(Ω)\{0} )

.

2 Local estimates for solutions of (1.1)–(1.3)

Theorem 2.1 Let (u, v)∈ D^1,p(R^N)× D^1,q(R^N) be a solution of(1.1)−(1.3) andτ= _N^N

−p,τ¯= _N^N

−q. Assume that max{p, q}< N,q≥pand

|f(x, u, v)| ≤C

|u|^p−1+|u|^p∗−1+|v|^q/p0+|v|^{(τ p)τ q0}

, (2.1)

and

|g(x, u, v)| ≤C

|v|^q−1+|v|^{τ q−1}+|u|^p/q0+|u|^{(τ q)0τ p}

, (2.2)

wherem⁰ is the conjugate of m andC is a constant. Then 1) For any R >0 andx∈R^N satisfying

Cmaxn

2^pSpτ^p−1,2^2q−pSq|B1|^q−pN R^q−pτ^q−1o

×

kuk^p(τ_Lp∗⁻(B¹⁾2R(x))+kvk^q(τ_Lqτ⁻(B¹⁾2R(x))

<1

(2.3)

where Sp andSq are the Sobolev constants, we have kukL^∞(B_R

2

(x))

≤ c(1 +R^q)

N(N−p) p3 maxn

R^p−Np kukL^p∗(B_R(x)), R^q−Np kvk^q/p_Lq∗(BR(x))

o . and

kvkL^∞(BR 2(x))

≤ c(1 +R^q)

N(N−p) qp2 maxn

R^q−Nq kvkL^q∗(BR(x)), R^p−Nq kuk

p q

L^p∗(B_R(x))

o . witch cindependent of u, v, xandR.

2) Moreover,

lim

|x|→+∞u(x) = lim

|x|→+∞v(x) = 0.

Remark 2.2 There exists an R0 such that for all R < R0, (2.3) is satisfied uniformly for all x∈Ω. This follows from the absolute continuity of the func- tionalsA7→R

A|u|^p∗dxandA7→R

A|v|^qτdx. To be more specific, for each >0 there exists η >0 such that for all R >0 and x∈R^N satisfying|BR(x)| ≤η, we have R

B_R(x)|u|^p∗dx < andR

B_R(x)|v|^qτdx < .

Proof Letx∈ R^N be fixed. For y ∈ B2R(x) and any function hdefined on B2R(x) we define

˜h(t) =h(y), t= y−x R . Since (u, v) is a solution for (1.1)–(1.3), then (˜u,˜v) satisfies

−∆_pu˜=R^pf(y,u,˜ v),˜ (2.4)

−∆qv˜=R^qg(y,u,˜ v).˜ (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∈ W0^1,p(B) kwk^p_Lpτ(B)≤Spk∇wk^p_Lp(B),

∀w∈ W0^1,q(B) kwk^q_Lqτ(B)≤2^q−p|B₁(0)|^q−pN S_qk∇wk^q_Lq(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

r0= 2, rn = 2− 1 σ

n−1

X

i=0

mi+p p

−1/p⁰

,

whereR is positive andσ=

∞

X

i=0

m_i+p p

−1/p⁰

. We denote byB_n =B(0, r_n) and we define η ∈ C0^∞(R^N) so that 0 ≤η ≤1, η = 1 in Bn+1, supp(η)⊂Bn

and

|∇η| ≤c

mn+p p

^1/p0

. (2.7)

We multiply (2.4) by|u˜|^mn˜uη^q, and integrate overBn. Using (2.1), we obtain I₁+I₂≤R^p(I₃+I₄+I₅+I₆), (2.8) where

I1= (1 +mn) Z

B_n

η^q|u˜|^mn|∇˜u|^pdx, I2=q

Z

Bn

η^q−1∇η.∇u˜|∇u˜|^p−2|˜u|^mnudx,˜ I3=C

Z

B_n

|u˜|^p+mnη^qdx, I4=C

Z

Bn

|u˜|^p∗+mnη^qdx, I₅=C

Z

B_n

|u˜|^mn˜u|v˜|^q/p0η^qdx, I6=C

Z

Bn

|u˜|^mn˜u|v˜|^{(τ p)τ q0}η^qdx.

Since 1 +mn =^(p−1)m_p ⁿ+^mn_p^+p, we deduce from Young inequality and the facts p≤q,|η| ≤1, that for anys >0

|I2| ≤ qs^p0 p⁰

m_n+p p

Z

B_n

η^q|∇u˜|^p|u˜|^mndx + q

ps^p

mn+p p

−p0^p Z

B_n

|∇η|^p|u˜|^mn+pdx

Choosingssuch that ^qsp

0

p⁰ ≤¹₂, and using (2.7), we have

|I₂| ≤ 1 2I₁+c

Z

B_n

|u˜|^mn+pdx. (2.9) We deduce from (2.8) and (2.9)

I1≤2R^p

6

X

i=3

Ii+c Z

Bn

|u˜|^mn+pdx. (2.10)

Using Sobolev inequality and observing that for anya≥0 andb≥0 (a+b)^p≤ 2^p−1(a^p+b^p), we have

η^q/pu˜^mn+pp

p

L^pτ(B_n)≤2^p−1Sp(I7+I8), (2.11) where

I7= (q p)^p

Z

Bn

η^q−p|∇η|^p|u˜|^mn+pdx≤c

mn+p p

^p−1Z

Bn

|u˜|^mn+pdx, and

I8=

mn+p p

^pZ

Bn

η^q|u˜|^mn|∇u˜|^pdx≤

mn+p p

^p−1

I1, thus we deduce from (2.10) that

η^q/pu˜^mn+pp

p

L^pτ(B_n)≤

m_n+p p

p−1

c Z

B_n

|u˜|^mn+pdx+ 2^pSpR^p

6

X

i=3

Ii

! . (2.12) First step. We construct the sequences (p_n)_n and (q_n)_n by

pn=pτⁿ, qn=qτⁿ, and we set

m_n=p(τⁿ−1), and l_n=q(τⁿ−1).

We show that if the condition Cmaxn

2^pSpR^pτ^n(p−1),2^2q−p|B1|^q−pN SqR^qτ^n(q−1)o

×

ku˜k^p(τ_Lp∗⁻(B¹⁾2)+k˜vk^q(τ_Lqτ⁻(B¹⁾₂)

< 1, is satisfied, the solution (˜u,v) belongs to˜ L^pn+1(Bn+1)×L^qn+1(Bn+1).

First, we start by estimating the integrals (Ii), i= 3, . . . ,6. We have I3=C

Z

Bn

|u˜|^p+mnη^qdx≤cku˜k^p_Lⁿpn(B_n). (2.13) Remarking that ^m_pⁿ⁺¹

n +

q p0

q_n = 1, we deduce from H¨older inequality that I5=C

Z

B_n

|u˜|^mnu˜|˜v|^q/p0η^qdx≤cku˜k^m_Lpnⁿ⁺¹(Bn)kv˜k^q/p_Lqn⁰(Bn). (2.14) We write mn+p^∗ = p(τ −1) +mn +p , q = τ q(^m_p^n+p

n+1). Observing that

m_n+p

pn+1 +^p(τ_p⁻_∗¹⁾ = 1,we deduce from H¨older inequality I4=C

Z

Bn

|u˜|^p∗+mnη^qdx≤C Z

Bn

|u˜|^p(τ−1)|u˜|^p+mnη^{τ q(}

mn+p pn+1)

dx

≤Ck˜uk^p(τ_Lp∗⁻(B¹⁾n)kη^q/pu˜^τnk^p_Lpτ(Bn).

(2.15)

Remark that τ q (τ p)⁰ − q

p⁰ =q(τ−1), q(τ−1)

τ q +mn+ 1 p_n+1 +

q p⁰

q_n+1 = 1, (2.16) and

τmn+ 1 pn+1

+τ

q p⁰

qn+1

= 1, then from H¨older inequality, we have

I₆=C Z

B_n

|u˜|^mnu˜|v˜|^{(τ p)τ q0}η^qdx

≤C Z

Bn

|v˜|^q(τ−1)η^{τ q(mn+1pn+1)}|u˜|^mn+1η^{τ q(}

q p0 qn+1)

|˜v|^q/p0dx

≤Ckv˜k^q(τ_Lτ q⁻(B¹⁾n)kη^q/pu˜^τnk^p(

1+mn pn )

L^pτ(Bn)kη˜v^τnk^q(

p0q qn) L^qτ(Bn).

(2.17)

Substitutingm_n byp(τⁿ−1) in (2.12), we obtain

kη^q/pu˜^τnk^p_Lpτ(B_n)−τ^n(p−1)2^pS_pR^p(I₄+I₆)

≤τ^n(p−1)

c Z

B_n

|u˜|^pndx+ 2^pSpR^p(I3+I5)

.

(2.18)

It follows from (2.13) - (2.17) and the factp≤qthat kη^q/pu˜^τnk^p_Lpτ(B_n)−C2^pSpR^pτ^n(p−1)

ku˜k^p(τ_Lp∗⁻(B¹⁾_n)kη^q/pu˜^τnk^p_Lpτ(B_n)

+kv˜k^q(τ_Lτ q⁻(B¹⁾_n)kη^q/pu˜^τnk^p

(1+mn) pn

L^pτ(B_n)kηv˜^τnk^q(

q p0 qn) L^qτ(B_n)

!

≤c(1 +R^q)τ^n(q−1)

ku˜k^p_Lⁿpn(B_n)+ku˜k^m_Lpnⁿ⁺¹(B_n)kv˜k^q/p_Lqn⁰(B_n)

.

(2.19)

Similarly, we have

kηv˜^τnk^q_Lqτ(B_n)−C2^2q−pS_q|B₁|^q−pN R^qτ^n(q−1)

k˜vk^q(τ_Lqτ⁻(B¹⁾_n)kηv˜^τnk^q_Lqτ(B_n)

+ku˜k^p(τ_Lτ p⁻(B¹⁾_n)kη˜v^τnk^q

(1+ln) qn

L^qτ(B_n)kη^q/pu˜^τnk^p(

p q0 pn) L^pτ(B_n)

≤c(1 +R^q)τ^n(q−1)k˜vk^q_Lⁿqn(B_n)+cR^qτ^n(q−1)k˜vk^l_Lⁿqn⁺¹(B_n)ku˜k^p/q_Lpn⁰(B_n).

(2.20)

Next, we defineθ_n+1= max{kη^q/pu˜^τnk^p_Lpτ(Bn),kηv˜^τnk^q_Lqτ(Bn)}, and

En = max{ku˜k^p_Lⁿpn(B_n),kv˜k^q_Lⁿqn(B_n)}^1/pn. Simple computations using H¨older inequality and the definition ofE_n andθ_n, show that

θn+1−Cmaxn

2^pSpR^pτ^n(p−1),2^2q−p|B1|^q−pN SqR^qτ^n(q−1)o

×

ku˜k^p(τ_Lp∗⁻(B¹⁾n)+kv˜k^q(τ_Lqτ⁻(B¹⁾n)

θn+1≤c(1 +R^q)τ^n(q−1)E_n^pn.

(2.21)

We know that there existsR0>0 such that for anyR < R0

Cmaxn

2^pS_pR^pτ^n(p−1),2^2q−p|B₂|^q−pN S_qR^qτ^n(q−1)o

×

ku˜k^p(τ_Lp∗⁻(B¹⁾₂)+kv˜k^q(τ_Lqτ⁻(B¹⁾₂)

<1.

(2.22)

Also, remark that

θ_n+1≥max{ku˜k^p_Lⁿpn+1(Bn+1),k˜vk^q_Lⁿqn+1(Bn+1)}

≥max{ku˜k^p_Lⁿ⁺¹pn+1(B_n+1),k˜vk^q_Lⁿ⁺¹qn+1(B_n+1)}^1/τ

=E_n+1^pn .

(2.23)

Therefore, from (2.21) - (2.23), and the factp≤q E_n+1^pn ≤c(1 +R^q)τ^n(q−1)E_n^pn. So

En+1≤(c(1 +R^q))^1/pnτ^n(q−1)pn En. This implies that

ku˜kL^pn+1(Bn+1)≤En+1≤(c(1 +R^q))

P∞ i=0

1 pτ i τ

P∞ i=0

i(q−1) pτ i E0. Since P_∞

i=0 1

pτⁱ = ^N_p2 andP_∞

i=0 i(q−1)

pτⁱ <∞, we deduce that ˜u∈L^pn+1(Bn+1).

Similarly, we have k˜vk^q/p_Lqn+1(Bn+1)≤ k˜vk

qn+1 pn+1

L^qn+1(Bn+1)≤E_n+1≤(c(1 +R^q))

P∞ i=0 1

pτ i τ

P∞ i=0

i(q−1) pτ i E₀, thereforev∈L^qn+1(Bn+1).

Second step We remark that hypothesis (2.3) is equivalent to Cmaxn

2^pSpR^pτ^p−1,2^2q−p|B1|^q−pN SqR^qτ^q−1o

ku˜k^p(τ_Lp∗⁻(B¹⁾₂)+kv˜k^q(τ_Lq∗⁻(B¹⁾₂)

<1.

We assume that R, u and v satisfy (2.3), which by the first step implies that (˜u,v)˜ ∈L^pτ2(B1)×L^qτ2(B1). We let δ= _τ2−^τ2τ+1 andχ = ^τ_δ. It is clear that 1< δ < τ, and soχ >1. We construct a sequences (s_n)_n and (t_n)_n by

s_n=pχⁿ, t_n=qχⁿ. In this step mn andrn are defined by

mn=p χⁿ

δ −1

, and

r0= 1, rn= 1− 1 2σ

n−1

X

i=0

mi+p p

−1/p⁰

,

which implies mn+p=sn/δ. Now, we estimate the integrals (Ii)i=3,...,6. We have

I3≤cku˜k^sn/δ

L^snδ (B_n)≤cku˜k^s_Lⁿsn^/δ(Bn). (2.24) Remarking that ^m_sⁿ⁺¹

n/δ +_t^q/p0

n/δ = 1, it follows from H¨older inequality that I₅≤cku˜k^mn+1

L^snδ (B_n)kv˜k^q/p0

L^tnδ (B_n)≤ ku˜k^m_Lsnⁿ⁺¹(Bn)kv˜k^q/p_Lsn⁰(Bn). (2.25) We have ^p(τ_pτ⁻₂¹⁾+^mn_s^+p

n = 1, thus from H¨older inequality we have

I₄≤cku˜k^p(τ_Lpτ⁻2¹⁾(Bn)ku˜k^s_Lⁿsn^/δ(Bn)≤cku˜k^s_Lⁿsn^/δ(Bn). (2.26) Observing that ^q(τ_qτ⁻₂¹⁾+^m_sⁿ⁺¹

n +^q/p_t ⁰

n = 1, it follows from H¨older inequality that I₆≤c

Z

B_n

|v˜|^q(τ−1)|u˜|^mn+1|˜v|^q/p0dx

≤ckv˜k^q(τ_Lqτ⁻2(B¹⁾_n)ku˜k^m_Lsnⁿ⁺¹(B_n)kv˜k^q/p_Ltn⁰(B_n)

≤cku˜k^m_Lsnⁿ⁺¹(B_n)kv˜k^q/p_Ltn⁰(B_n).

(2.27)

We deduce from (2.12), (2.24)–(2.27) and the factp≤qthat

η^q/pu˜^χn/δ

p

L^pτ(Bn)≤cχ^n(q−1)(1 +R^q)

ku˜k^s_Lⁿsn^/δ(B_n)+ku˜k^m_Lsnⁿ⁺¹(B_n)kv˜k^q/p_Ltn⁰(B_n)

(2.28) Similarly, we have

ηv˜^χn/δ

q

L^qτ(Bn)≤cχ^n(q−1)(1 +R^q)

kv˜k^t_Lⁿtn^/δ(B_n)+kv˜k^l_Lⁿtn⁺¹(B_n)ku˜k^p/q_Lsn⁰(B_n)

(2.29) As in the first step, we let Λ_n= maxn

ku˜k^s_Lⁿsn(B_n),k˜vk^t_Lⁿtn(B_n)

o1/sn

Γn = maxn

kη^q/pu˜^χn/δk^p_Lpτ(Bn),kηv˜^χn/δk^q_Lqτ(Bn)

o and Υ_n= maxn

ku˜k^s_Lⁿsn(B_n),kv˜k^t_Lⁿtn(B_n)

o_tn¹

. Simple computations show that ku˜k^m_Lsnⁿ⁺¹(Bn)k˜vk^q/p_Ltn⁰(Bn)≤minn

Λ^s_n^n/δ,Υ^t_n^n/δo

, (2.30)

and

kv˜k^l_Lⁿtn⁺¹(Bn)ku˜k^p/q_Lsn⁰(Bn)≤minn

Λ^s_n^n/δ,Υ^t_n^n/δo

. (2.31)

Also, remark that Γn≥maxn

ku˜k^s_Lⁿsn+1^/δ (Bn+1),kv˜k^t_Lⁿtn+1^/δ (B_n+1)

o

= Λ^s_n+1^n/δ= Υ^t_n^n/δ. (2.32)

Thus, we deduce from (2.28)–(2.32) that

Λ^s_n+1^n/δ≤cχ^n(q−1)(1 +R^q) Λ^s_n^n/δ, and so

Λn+1≤c^δ/snχ^n(q−1)δsn (1 +R^q)^δ/snΛn. Which implies that

ku˜kL^sn(Bn)≤Λn≤c

P_∞

i=0 δ siχ

P_∞

i=0 i(q−1)δ

si (1 +R^q)

P_∞

i=0 δ si Λ0. SinceP_∞

i=0 δ

si =_p(τ^δτ_−δ),andP_∞

i=0 i(q−1)δ

si <∞,then ku˜kL^∞(B1

2

)≤ lim

n→+∞supku˜kL^sn(Bn)

≤c(1 +R^q)^{p(τ−δ)δτ} maxn

ku˜kL^p(B1),k˜vk^q/p_Lq(B₁)

o . Similarly, we have

Υn+1≤c^tnδ χ^n(q−1)δtn (1 +R^q)^tnδ Υn

Asntends to infinity, we obtain kv˜kL^∞(B₁

2

)≤ lim

n→+∞supk˜vkLtn(B_n)

≤c(1 +R^q)^{q(τ−δ)δτ} maxn

k˜vkL^p(B₁),ku˜k

p q

L^q(B1)

o . By the imbeddings

L^p∗(B₁)⊂L^p(B₁) and L^q∗(B₁)⊂L^q(B₁), and the fact

δτ

τ−δ = τ

(τ−1)² =N(N−p) p² , we have

ku˜kL^∞(B₁

2

)≤c(1 +R^q)

N(N−p) p3 maxn

ku˜kL^p∗(B₁),kv˜k^q/p_Lq∗(B1)

o , and

k˜vkL^∞(B₁

2

)≤c(1 +R^q)^N(N−p)qp2 maxn

k˜vkL^p∗(B₁),ku˜k

p q

L^q∗(B₁)

o. Coming back to (u, v) by a simple change of variables, we find

kukL^∞(B_R

2(x))

≤ c(1 +R^q)

N(N−p) p3 maxn

R^p−Np kukL^p∗(B_R(x)), R^q−Np kvk^q/p_Lq∗(BR(x))

o .

and

kvkL^∞(B_R

2(x))

≤ c(1 +R^q)

N(N−p) qp2 max

R^q−Nq kvkL^q∗(B_R(x)), R^p−Nq kuk

p q

L^p∗(BR(x))

o . The proof of 2) follows from 1) and Remark 2.2 ♦ Proposition 2.3 Let (u, v)∈ D^1,p(R^N)× D^1,q(R^N)a solution of (1.1)–(1.3).

We assumeq≥p,

|f(x, u, v)| ≤C

|u|^p−1+|v|^q/p0+ 1

, (2.33)

and

|g(x, u, v)| ≤C

|v|^q−1+|u|^p/q0+ 1

, (2.34)

wherem⁰ is the conjugate ofm. Then kukL^∞(B₁)≤c(1 +R^q)^pN2 maxn

1, R^p−Np kukL^p∗(B2), R^q−Np kvk^q/p_Lq∗(B₂)

o

, (2.35) and

kvkL^∞(B₁)≤c(1 +R^q)^pqN maxn

1, R^p−Nq kuk

p q

L^p∗(B₂), R^q−Nq kvkL^q∗(B₂)

o. (2.36) Proof We use the same change of variables as in the proof of Theorem 2.1.

Thus, we obtain that (˜u,˜v) satisfies (2.4) and (2.5). Also we keep the same sequences (mn)n, (rn)n, (Bn)nand the same functionη. We multiply Equation (2.4) by|u˜|^mnuη˜ ^q, and integrate overBn. Using (2.33), we have

I1+I2≤R^p(I3+I4+I5), (2.37) where

I1= (1 +mn) Z

Bn

η^q|u˜|^mn|∇˜u|^pdx, I2=q

Z

B_n

η^q−1∇η.∇u˜|∇u˜|^p−2|˜u|^mnudx,˜ I3=C

Z

Bn

|u˜|^p+mnη^qdx, I₄=C

Z

B_n

|u˜|^mn˜u|v˜|^q/p0η^qdx, I5=C

Z

Bn

|u˜|^mn˜uη^qdx.

The integralsI1, I2, I3 and I4 are the same to those obtained in Theorem 2.1.

Simple computations used before show that

η^q/pu˜^mn+pp

p

L^pτ(B_n)≤

mn+p p

^p−1

c Z

Bn

|u˜|^mn+pdx+ 2^pSpR^p

5

X

i=3

Ii

! . (2.38)

Now, we define (pn)n and (qn)n by

pn=pτⁿ, qn=qτⁿ,

and let mn =p(τⁿ−1),and , ln =q(τⁿ−1). Then we estimate the integrals Ii, i= 3, . . . ,5. It is clear from (2.13) and (2.14) that

I3≤cku˜k^p_Lⁿpn(B_n) and I4≤cku˜k^m_Lpnⁿ⁺¹(B_n)k˜vk^q/p_Lqn⁰(B_n). (2.39) On the other hand

I5≤C Z

B_n

|u˜|^mn+1dx=cku˜k^m_Lmn+1ⁿ⁺¹(Bn)≤c|Bn|^{(mn1 −pn1 )(mn+1)}ku˜k^m_Lpnⁿ⁺¹(Bn)

≤c|B2|^{p−1pτ n}ku˜k^m_Lpnⁿ⁺¹(B_n)

≤cku˜k^m_Lpnⁿ⁺¹(B_n).

(2.40) We deduce from (2.38)–(2.40) that

ku˜k^p_Lⁿpn+1(B_n+1)≤ kη^q/pu˜^τnk^p_Lpτ(Bn)

≤cτ^n(p−1)

ku˜k^p_Lⁿpn(B_n)

+R^p

ku˜k^p_Lⁿpn(Bn)+ku˜k^m_Lpnⁿ⁺¹(Bn)k˜vk^q/p_Lqn⁰(Bn)+ku˜k^m_Lpnⁿ⁺¹(Bn) . (2.41) Similarly, we have

k˜vk^q_Lⁿqn+1(Bn+1)≤ kη˜v^τnk^q_Lqτ(Bn)

≤cτ^n(q−1)

k˜vk^q_Lⁿqn(Bn)

+R^q

kv˜k^q_Lⁿqn(Bn)+kv˜k^l_Lⁿqn⁺¹(Bn)ku˜k^p/q_Lpn⁰(Bn)+kv˜k^l_Lⁿqn⁺¹(Bn) . (2.42) Following the proof of Theorem 2.1 we let

E_n = maxn

1,ku˜_nk^p_Lⁿpn(B_n),k˜v_nk^q_Lⁿqn(B_n)

o1/pn

and Fn=n

1,k˜unk^p_Lⁿpn(B_n),kv˜nk^q_Lⁿqn(B_n)

o_qn¹

. We obtain ku˜kL^∞(B1)≤ lim

n→+∞supku˜kL^pn(Bn)≤En

≤c(1 +R^q)^pN2 E0

=c(1 +R^q)

N p2 maxn

1,ku˜kL^p(B₂),kv˜k^q/p_Lq(B2)

o .

(2.43)

k˜vkL^∞(B1)≤ lim

n→+∞supkv˜kL^qn(Bn)≤Fn

≤c(1 +R^q)^pqN F₀

=c(1 +R^q)^pqN maxn 1,ku˜k

p q

L^p(B₂),kv˜kL^q(B2)

o .

(2.44)

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)∈ D^1,p(Ω)× D^1,q(Ω)a solution of (1.1)–(1.3). We assume that there exist a functionsa, b∈L¹(Ω)∩L^∞(Ω)and a constantC such that

|f(x, u, v)| ≤a(x) +C(|u|^p−1+|v|^q/p0), (3.1)

|g(x, u, v)| ≤b(x) +C(|v|^q−1+|v|^p/q0), (3.2) wherep >1,q >1. Then

1)(u, v)∈L^σ(Ω)×L^η(Ω) for all(σ, η)∈[p^∗,+∞)×[q^∗,+∞).

2) lim

|x|→+∞u(x) = lim

|x|→+∞v(x) = 0.

Proof 1) Let pn = pτⁿ, qn = qτⁿ, mn = τⁿ −1, tn = τⁿ −1, Tk(u) = max{−k,min{k, u}} and w = |Tk(u)|^pmnTk(u), with k > 0. Multiplying the equation (1.1) byw and integrating over Ω, we obtain

(pmn+ 1) Z

Ω

|∇Tk(u)|^p|Tk(u)|^pmndx= Z

Ω

f(x, u, v)w dx.

Observing that ( 1

m_n+ 1)^p|∇(Tk(u))^mn+1|^p=Tk(u)^pmn|∇Tk(u)|^p, (3.3) we deduce from H¨older and Sobolev inequalities that for any 0 < γ < 1, we have

Z

Ω

|Tk(u)|^τ(pmn+p))^1/τ

≤c

kak¹_∞^−γkak^γ_L1(Ω)kuk^pm_Lpnⁿ(Ω)⁺¹+kuk^p_Lⁿpn(Ω)+kvk^q/p_Lqn⁰(Ω)kuk^m_Lpnⁿ⁺¹(Ω)

.

(3.4)

withcdepending from n. Lettingktend to infinity in (3.4), we obtain kuk^p_Lⁿpn+1(Ω)≤c

kuk^pm_Lpnⁿ(Ω)⁺¹+kuk^p_Lⁿpn(Ω)+kvk^q/p_Lqn⁰(Ω)kuk^m_Lpnⁿ⁺¹(Ω)

. (3.5) We derive from (3.5) that u∈L^pn(Ω) for all n∈N. Similarly, we prove that v ∈ L^qn(Ω) for all n∈ N. By interpolation inequality (see [3]) we prove that