Nonlinear Analysis 75 (2012) 3099–3106

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

The spatial behavior of a strongly coupled non-autonomouselliptic system

Ling Zhou a,∗, Shan Zhang a, Zuhan Liu a,b, Zhigui Lin a

a School of Mathematical Science, Yangzhou University, Yangzhou 225002, Chinab School of Mathematical Science, Xuzhou Normal University, Xuzhou, 221116, China

a r t i c l e i n f o

Article history:Received 14 July 2011Accepted 8 December 2011Communicated by Enzo Mitidieri

MSC:35J5535Q40

Keywords:Strongly coupledCompetition systemSpatial segregation

a b s t r a c t

This article is concerned with the spatial behavior of a strongly coupled non-autonomouselliptic system modeling the steady state of populations that compete in some region. Asthe competition rate tends to infinity, we obtain the uniform convergence result and provethat non-negative solution of the system converges to the positive and negative parts of asolution of a scalar limit problem.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, the study of spatial behavior of interacting species has developed as an exciting problem in populationecology. Different kinds of mathematical models are used to investigate phenomena of coexistence and exclusion ofcompeting species [1–8]. According to the Gause principle of competitive exclusion, two competing species cannot coexistunder strong competition. The migration or the spatial distribution changes the situation and then species can coexist dueto the segregation of their habitats.

Among many models proposed so far, the elliptic systems of the form1u + f (u) − kuv = 0, x ∈ Ω,1v + g(v) − kuv = 0, x ∈ Ω,u(x) = v(x) = 0, x ∈ ∂Ω,

(1.1)

are used to study the steady states of populations with densities u and v that compete in a region Ω . The functions f and gare positive on (0,M) for someM > 0 and negative otherwise. The parameter k > 0 is the interspecies competition rate.

Dancer and Du [9] and related works studied the large interaction limit as k → ∞ of this system. They proved thatk-dependent non-negative solutions (uk, vk) converge to the positive and negative parts respectively of a limit function wsatisfying the equation

1w + f (w+) − g(−w−) = 0, x ∈ Ω, (1.2)

Theworkwas partially supported by PRC grants NSFC 11071206, NSFC 11071209, the Tian Yuan Fund ofMathematics (China) 11126045, and YangZhouUniversity Fund 2011CXJ006.∗ Corresponding author.

E-mail address: zhouling53@yahoo.com.cn (L. Zhou).

0362-546X/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2011.12.008

3100 L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106

where w+:= max(0, w),w−

:= min(0, w) and w = w++ w−. This yields spatial segregation of the densities u and v.

Here we note that the linear combination wk= uk

− vk satisfies the equation

1wk+ f (uk) − g(vk) = 0, x ∈ Ω. (1.3)

Crooks and Dancer [10] discussed the k → ∞ limit of several systems with more general competitive interaction terms.One of these is the non-autonomous system

1u + f (u) − α1(x)kuv = 0, x ∈ Ω,1v + g(v) − α2(x)kuv = 0, x ∈ Ω,u(x) = v(x) = 0, x ∈ ∂Ω.

(1.4)

They established some improved convergence results which are of great importance for dealing with the limit problem.They showed that as k tends to infinity, non-negative solutions of the system converge to the positive and negative parts ofa solution of a limit problem.

We consider the following strongly coupled non-autonomous system of elliptic equations:∆[(d1 + β11uk

1 + β12uk2)u

k1] + uk

1(a1 − b1uk1) − kα1(x)uk

1uk2 = 0, x ∈ Ω,

∆[(d2 + β21uk1 + β22uk

2)uk2] + uk

2(a2 − b2uk2) − kα2(x)uk

1uk2 = 0, x ∈ Ω,

uk1(x) = uk

2(x) = 0, x ∈ ∂Ω,

(1.5)

where

α1, α2 ∈ C2 (Ω, [α0, +∞)) for some constant α0 > 0, (1.6)

such that α1 and α2 are bounded below by a strictly positive constant α0. di, ai, bi (i = 1, 2) and k are all positive constantswhile βij (i, j = 1, 2) denote non-negative constants. The k-dependent solution pair (uk

1, uk2) represents the densities of two

competing species, d1 and d2 are their diffusion rates, a1 and a2 denote the intrinsic growth rates, b1 and b2 account forintraspecies competitions, and kα1 and kα2 are the interspecies competitions. β11 and β22 are usually referred to as self-diffusion pressures, and β12 and β21 are cross-diffusion pressures. Throughout the paper, we will always suppose that k ∈ Nand Ω is a bounded, open connected subset of RN with smooth boundary ∂Ω .

System (1.5) represents a model of two competing species with self-pressures and cross-population pressures. In thecase where βij = 0 for i, j = 1, 2, system (1.5) is the classic Lotka–Volterra competition model, while if either β12 = 0 orβ21 = 0 the system becomes strongly coupled. This model has been investigated by many workers [6,11–17], and variousexisting results for system (1.5) have been developed. In particular, the work by Lou and Ni [14] characterized the existenceof nonconstant positive solutions for both the small and the large competition cases, while those in [12,13] were concernedwith the existence of positive solutions in relation to a pair of curves in the (a1, a2)-plane for both large and small cross-diffusion cases. For the existing results concerning the time-dependent case of System (1.5), we refer the reader to [18,19]and references therein.

In this paper, we study the large competition limit of the strongly coupled elliptic system (1.5). First, we give some prioriestimates of (1.5). Second, we prove the spatial segregation conclusion and give some improved convergence results (seeLemmas 3.5 and 3.6). Last, we prove that non-negative solutions of the system converge to the positive and negative partsof a solution of a scalar limit problem.

The rest of the paper is organized as follows. Section 2 is devoted to giving the priori estimates. Section 3 deals with theconvergence properties as the competition rate tends to infinity. In Section 4, we characterize the limiting problem.

2. The priori estimates

We begin with some basic priori bounds.

Lemma 2.1. Let g ∈ C(Ω), w ∈ C2(Ω) ∩ C1(Ω) satisfy1w(x) + g(x, w(x)) ≥ 0, x ∈ Ω,w(x) = 0, x ∈ ∂Ω.

If w(x0) = maxΩ w > 0, then g(x0, w(x0)) ≥ 0.

Proof. If w(x0) = maxΩ w > 0, we have x0 ∈ Ω, ∇w(x0) = 0 and 1w(x0) ≤ 0. It is clear that g(x0, w(x0)) ≥ −1w(x0)≥ 0.

Theorem 2.2. Define m1 and m2 as

m1 :=a1b1

1 +

β11

d1·a1b1

+β12

d1·a1α0

, m2 :=

a2b2

1 +

β22

d2·a2b2

+β21

d2·a2α0

.

L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106 3101

Let (uk1, u

k2) be a non-negative solution of the problem (1.5) for some k ≥ 1. Then

0 ≤ uk1 ≤ m1 and 0 ≤ uk

2 ≤ m2.

Proof. We will only prove the priori estimate of uk1; the proof of uk

2 is similar.Let δ(x) = uk

1(d1 + β11uk1 + β12uk

2); then1δ + uk

1(a1 − b1uk1) − kα1(x)uk

1uk2 = 0, x ∈ Ω,

δ(x) = 0, x ∈ ∂Ω.

If maxΩ δ(x) = 0, then uk1(x) ≡ 0. If there exists x0 ∈ Ω such that δ(x0) = maxΩ δ(x) > 0, then uk

1(x0) > 0. UsingLemma 2.1, we have a1 − b1uk

1(x0) − kα1(x0)uk2(x0) ≥ 0. Combining (1.6) and k ≥ 1 yields that

uk1(x0) ≤

a1b1

and uk2(x0) ≤

a1kα1(x0)

≤a1α0

.

From the definition of δ(x), we haved1 + β11 max

Ω

uk1

max

Ω

uk1 ≤ max

Ω

δ(x) ≤a1b1

d1 + β11 ·

a1b1

+ β12 ·a1α0

.

This implies that for all k ≥ 1,

maxΩ

uk1 ≤

a1b1

1 +

β11

d1·a1b1

+β12

d1·a1α0

= m1,

which completes the proof of Theorem 2.2.

Define

zk1 := uk1(d1 + β11uk

1 + β12uk2), zk2 := uk

2(d2 + β21uk1 + β22uk

2), (2.1)

and

F := uk1(d1 + β11uk

1 + β12uk2) − zk1, G := uk

2(d2 + β21uk1 + β22uk

2) − zk2.

If the Jacobian determinant

J =∂(F ,G)

∂(uk1, u

k2)

=

d1 + 2β11uk1 + β12uk

2 β12uk1

β21uk2 d2 + β21uk

1 + 2β22uk2

> d1d2 > 0,

then there exist inverse functions uk1 = f1(zk1, z

k2) and uk

2 = f2(zk1, zk2), which are continuous and have continuous partial

derivatives. So the strongly coupled problem (1.5) can be changed to1zk1 + f1(zk1, z

k2)[a1 − b1f1(zk1, z

k2)] − kα1(x)f1(zk1, z

k2)f2(z

k1, z

k2) = 0, x ∈ Ω,

1zk2 + f2(zk1, zk2)[a2 − b2f2(zk1, z

k2)] − kα2(x)f1(zk1, z

k2)f2(z

k1, z

k2) = 0, x ∈ Ω,

zk1(x) = zk2(x) = 0, x ∈ ∂Ω.

(2.2)

Combining Theorem 2.2 and the definition (2.1), the following theorem can be obtained naturally.

Theorem 2.3. Let (uk1, u

k2) be a non-negative solution of the problem (1.5) for some k ≥ 1, and (zk1, z

k2) be defined as (2.1). Define

M1,M2 as

M1 := (d1 + β11m1 + β12m2)m1, M2 := (d2 + β21m1 + β22m2)m2,

where m1,m2 be as in Theorem 2.2. Then:

1. (zk1, zk2) is a non-negative solution of the problem (2.2), and

0 ≤ zk1 ≤ M1, 0 ≤ zk2 ≤ M2 in Ω. (2.3)

2. uki (x) = 0 if and only if zki (x) = 0, for i = 1, 2.

3. f1(zk1, zk2) > 0 and f2(zk1, z

k2) > 0 if zk1 > 0 and zk2 > 0.

3. Convergence properties as k → ∞

In this section, we establish some convergence results as k → ∞. Thesewill be used to deal with the limit of the problem(1.5).

3102 L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106

Lemma 3.1. There exist positive constants K1 and K2, which are independent of k, such that if (uk1, u

k2) is a non-negative solution

of (1.5) for some k ≥ 1, and (zk1, zk2) is defined by (2.1), then

Ω

|∇zki (x)|2dx ≤ Ki for i = 1, 2. (3.1)

Proof. Multiplication of Eq. (2.2) containing 1zk1 by zk1 and integration over Ω yields that

−

Ω

|∇zk1(x)|2dx +

Ω

f1(a1 − b1f1)zk1 = −

Ω

|∇zk1(x)|2dx +

Ω

uk1(a1 − b1uk

1)zk1 ≥ 0,

since zk1(x) = 0 for all x ∈ ∂Ω and f1, f2 are non-negative. The estimate for∇zk1 then follows from Theorems 2.2 and 2.3.

Combining Theorem 2.3 and Lemma 3.1, we have the following, Lemma 3.2.

Lemma 3.2. Given a sequence of non-negative solutions (uk1, u

k2)k∈N of the problem (1.5) and letting (zk1, z

k2) be as in the statement

in Theorem 2.3, then there exist subsequences zkn1 , zkn2 and non-negative functions z1, z2 ∈ L∞(Ω) ∩ W 1,2(Ω) such that

zkn1 z1, zkn2 z2 in W 1,2(Ω),

and

zkn1 → z1, zkn2 → z2 in L2(Ω) and a.e. in Ω as kn → ∞.

To prove the spatial segregation of uk and vk as k tends to infinity, we give the following estimates.

Lemma 3.3. Define ϕ as the first eigenfunction of the operator −∆ in Ω . There exists a positive constant C1, independent of k,such that if (uk

1, uk2) is a non-negative solution of (1.5) for some k ≥ 1, and (zk1, z

k2) is defined as in (2.1), then

k

Ω

f1(zk1, zk2)f2(z

k1, z

k2)ϕdx ≤ C1,

that is

k

Ω

uk1u

k2ϕdx ≤ C1.

Proof. Since ϕ is the first eigenfunction of the operator −∆ in Ω , the function ϕ satisfies ∥ϕ∥H10 (Ω) = 1 and

−1ϕ = λϕ, x ∈ Ω,ϕ = 0, x ∈ ∂Ω,

with λ > 0 and ϕ > 0 in Ω . Integrating the equation for zk1 in (2.2) over Ω after multiplication by ϕ yields

k

Ω

α1f1(zk1, zk2)f2(z

k1, z

k2)ϕdx =

Ω

1zk1ϕdx +

Ω

f1(zk1, zk2)[a1 − b1f1(zk1, z

k2)]ϕdx

=

∂Ω

∂zk1∂ν

ϕ − zk1∂ϕ

∂ν

ds +

Ω

zk11ϕdx +

Ω

f1(zk1, zk2)[a1 − b1f1(zk1, z

k2)]ϕdx

=

Ω

uk1(a1 − b1uk

1)ϕdx,

since ϕ = zk1 = 0 on ∂Ω . Combining Theorem 2.2 and α1(x) ≥ α0 > 0 in Ω , we get the conclusion.

Then we give the basic segregation result.

Lemma 3.4. Let z1 and z2 be as in the statement in Lemma 3.2, and

u1 := f1(z1, z2), u2 := f2(z1, z2).

Then

u1u2 = 0 a.e. in Ω,

that is

z1z2 = 0 a.e. in Ω.

Proof. It is a consequence of Lemmas 3.2 and 3.3.

L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106 3103

Inspired by the work of Crooks and Dancer [10], we have the following two lemmas.

Lemma 3.5. Let zkn1 , zkn2 , z1 and z2 be as in the statement of Lemma 3.2. Then

∇zkn1 → ∇z1, ∇zkn2 → ∇z2 in L2(Ω),

and hence

zkn1 → z1, zkn2 → z2 in W 1,2(Ω).

Proof. By Lemma 3.2, it suffices to show that as kn → ∞,Ω

|∇zkn1 |2dx →

Ω

|∇z1|2dx.

Weak lower-semicontinuity of the norm implies thatΩ

|∇z1|2dx ≤ lim infkn→∞

Ω

|∇zkn1 |2dx.

It remains to establish thatΩ

|∇z1|2dx ≥ lim supkn→∞

Ω

|∇zkn1 |2dx.

First note that it follows from Lemma 3.4, and Lemma 7.7 in [20], that

∇z1 · ∇z2 = 0 a.e. in Ω. (3.2)

Then multiplying the expression in (2.2) containing 1zkn2 by the limit z1 and integrating over Ω yields

−

Ω

∇z1 · ∇zkn2 dx +

Ω

f2(zkn1 , zkn2 )(a2 − b2f2(z

kn1 , zkn2 ))z1dx − kn

Ω

α2z1f1(zkn1 , zkn2 )f2(z

kn1 , zkn2 )dx = 0.

Lemmas 3.2 and 3.4, together with (3.2), give that, as kn → ∞,Ω

∇z1 · ∇zkn2 dx →

Ω

∇z1 · ∇z2dx = 0,Ω

f2(zkn1 , zkn2 )(a2 − b2f2(z

kn1 , zkn2 ))z1dx →

Ω

u2(a2 − b2u2)z1dx = 0.

So we have

kn

Ω

α2z1f1(zkn1 , zkn2 )f2(z

kn1 , zkn2 )dx = kn

Ω

α2z1ukn1 ukn

2 dx → 0 as kn → ∞. (3.3)

Since α1(x), α2(x) ≥ α0 > 0 in Ω , we have

kn

Ω

α1z1f1(zkn1 , zkn2 )f2(z

kn1 , zkn2 )dx = kn

Ω

α1z1ukn1 ukn

2 dx → 0 as kn → ∞. (3.4)

Now multiplying the expression in (2.2) containing 1zkn1 by the limit z1 and integrating over Ω yields

−

Ω

∇z1 · ∇zkn1 dx +

Ω

f1(zkn1 , zkn2 )(a1 − b1f1(z

kn1 , zkn2 ))z1dx − kn

Ω

α1z1f1(zkn1 , zkn2 )f2(z

kn1 , zkn2 )dx = 0,

and then passing to the limit as kn → ∞, using Lemma 3.2 and (3.4), yields thatΩ

|∇z1|2dx =

Ω

z1u1(a1 − b1u1)dx. (3.5)

Finally, multiplication of the expression in (2.2) containing 1zkn1 by zkn1 and integration over Ω gives

−

Ω

|∇zkn1 |2dx +

Ω

ukn1 (a1 − b1u

kn1 )zkn1 dx − kn

Ω

α1zkn1 f1(z

kn1 , zkn2 )f2(z

kn1 , zkn2 )dx = 0.

Since f1(zkn1 , zkn2 ) = ukn

1 , f2(zkn1 , zkn2 ) = ukn

2 and zkn1 are non-negative, the above equality implies thatΩ

|∇zkn1 |2dx ≤

Ω

ukn1 (a1 − b1u

kn1 )zkn1 dx →

Ω

z1u1(a1 − b1u1)dx, (3.6)

which, together with (3.5), gives the result.

3104 L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106

Lemma 3.6. Let ε > 0. There exists k0 ∈ N such that if k ≥ k0 and (zk1, zk2) is as in the statement of Lemma 3.2, we have

zk1(x) ≤ ε or zk2(x) ≤ ε. (3.7)

Proof. Suppose, for a contradiction, that there exist ε0 > 0 and sequences kj → ∞, xkj ∈ Ω such that

zkj1 (xkj) ≥ ε0 and z

kj2 (xkj) ≥ ε0.

For each j, define x′=kj(x − xkj) and a set Ωj such that x′

∈ Ωj whenever x ∈ Ω . Then for x′∈ Ωj, the functions Z

kj1 , Z

kj2

defined by Zkji (x′) = z

kji (x), i = 1, 2, satisfy

1Zkj1 + k−1

j f1(Zkj1 , Z

kj2 )a1 − b1f1(Z

kj1 , Z

kj2 )− α1

xkj +

x′kj

f1(Z

kj1 , Z

kj2 ) · f2(Z

kj1 , Z

kj2 ) = 0 in Ωj,

1Zkj2 + k−1

j f2(Zkj1 , Z

kj2 )a2 − b2f2(Z

kj1 , Z

kj2 )− α2

xkj +

x′kj

f1(Z

kj1 , Z

kj2 ) · f2(Z

kj1 , Z

kj2 ) = 0 in Ωj,

Zkj1 = Z

kj2 = 0 on ∂Ωj,

(3.8)

and 0 ∈ Ωj. Zkj1 (0) > ε0, Z

kj2 (0) > ε0.

We can easily see that xkj is bounded away from ∂Ω independently of j, fromwhich it follows both that dist(0, ∂Ωj) → ∞

as j → ∞, and thatwe can assume,without loss of generality, that there exists x ∈ Ω such that xj → x as j → ∞. Then givenan arbitrary compact set K ⊂ RN , K ⊂ Ωj for j sufficiently large, and it follows immediately from (3.8) and Theorem 2.3that 1Z

kji (i = 1, 2) are bounded in L∞(Ωj), independently of j, and so Z

kji are bounded in W 2,p(K) for every p ∈ [1, ∞)

and thus in C1,α(K) for each α ∈ (0, 1). Hence given α ∈ (0, 1), there are subsequences, not relabeled, of Zkji (i = 1, 2) that

converge strongly in C1,α(K) for each compact set K ⊂ RN to limit functions Z1, Z2 ∈ C1,α(RN) which satisfy the weak formof the system:

1Z1 − α1(x)f1(Z1, Z2)f2(Z1, Z2) = 0 in RN ,

1Z2 − α2(x)f1(Z1, Z2)f2(Z1, Z2) = 0 in RN .(3.9)

Wenote that (Z1, Z2) ∈ (C1,α(RN))2 and is a classical solution of (3.9). Using standard regularity theory and a diagonalizationargument, we show that a subsequence of (Z

kj1 , Z

kj2 ), again not relabeled, converges uniformly on all compact subsets of RN

to a solution (Z1, Z2) of (3.9), with

0 6 Z1 6 M1, 0 6 Z2 6 M2, Z1(0) > ε0, Z2(0) > ε0, (3.10)

whereM1,M2 are as in Theorem 2.3.Now it follows from (3.9) and (3.10) that

1Zi > 0, Zi > 0, and Zi is bounded on RN , (i = 1, 2).

Hence Z1(x) tends to sup Z1 as |x| → ∞ along almost all directions in the unit sphere ([21, Thm 3.21]), and likewise, Z2(x)tends to sup Z2 as |x| → ∞ along almost all directions. In particular, there is a direction along which both Z1 → sup Z1 andZ2 → sup Z2, as |x| → ∞. Let xnn∈N be a sequence of points along this direction such that |xn| → ∞ as n → ∞. Then localestimates similar to those described above for (Z

kj1 , Z

kj2 ) imply that a subsequence of the function pairs (Z1(·+xn), Z2(·+xn))

converges to a solution (Z1, Z2) of system (3.9) with the property that

Z1(0) = limn→∞

Z1(xn) = sup Z1 = supZ1,Z2(0) = limn→∞

Z2(xn) = sup Z2 = supZ2. (3.11)

It is then immediate from (3.11) that 1Z1(0) 6 0. ButZ1(0) = sup Z1 > Z1(0) > ε0 andZ2(0) = sup Z2 > Z2(0) > ε0. So(3.9) implies that

1Z1(0) = α1(x)f1(Z1(0),Z2(0)) · f2(Z1(0),Z2(0)) > 0.

Then the lemma is proved.

L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106 3105

4. Characterization of the k → ∞ limit

Say we are given a sequence of non-negative solutions (uk1, u

k2)k∈N of the problem (1.5). (zk1, zk2)k∈N , where (zk1, zk2) is

defined as in (2.1), is a sequence of non-negative solutions of the problem (2.2). Define

wk:= α2zk1 − α1zk2.

Then wk satisfies the equation1wk= zk11α2 − zk21α1 + 2∇α2 · ∇zk1 − 2∇α1 · ∇zk2

− α2f1(zk1, zk2)(a1 − b1f1(zk1, z

k2)) + α1f2(zk1, z

k2)(a2 − b2f2(zk1, z

k2)) in Ω,

wk= 0 on ∂Ω.

(4.1)

Theorem 4.1. Let zkn1 , zkn2 and z1, z2 be as in the statement of Lemma 3.2, and define

w := α2z1 − α1z2. (4.2)

Then:(i) (wkn)+ − α2z

kn1 → 0 and (wkn)− + α1z

kn2 → 0 in C(Ω) as kn → ∞;

(ii) wkn is bounded in W 2,2(Ω) ∩ W 1,20 (Ω) ∩ C(Ω) independently of kn;

(iii) wkn → w in W 1,20 (Ω) as kn → ∞, and

w+= α2z1 = α2u1(d1 + β11u1), w−

= −α1z2 = −α1u2(d2 + β22u2);

(iv) (the limit equation) the function w is a weak solution of the problem1w = α−1

2 w+1α2 + α−11 w−1α1 + 2∇α2 · ∇(α−1

2 w+) + 2∇α1 · ∇(α−11 w−)

− α2f1(α−12 w+, −α−1

1 w−)a1 − b1f1(α−1

2 w+, −α−11 w−)

+ α1f2(α−1

2 w+, −α−11 w−)

a2 − b2f2(α−1

2 w+, −α−11 w−)

in Ω,

w = 0 on ∂Ω,

(4.3)

in the sense that w ∈ W 1,20 (Ω) is such that for all φ ∈ W 1,2

0 (Ω),

−

Ω

∇w · ∇φdx =

Ω

α−12 w+1α2 + α−1

1 w−1α1 + 2∇α2 · ∇(α−12 w+) + 2∇α1 · ∇(α−1

1 w−)

− α2f1(α−12 w+, −α−1

1 w−)a1 − b1f1(α−1

2 w+, −α−11 w−)

+ α1f2(α−1

2 w+, −α−11 w−)

a2 − b2f2(α−1

2 w+, −α−11 w−)

φdx; (4.4)

(v) (regularity) w ∈ W 2,p(Ω) ∩ C1,η(Ω) for all p ∈ [1, ∞) and η ∈ (0, 1), and satisfies ((4.3)) pointwise a.e. in Ω .

Proof. (i) Taking ε > 0 and applying Lemma 3.6 yields the existence of k0 such that for each x ∈ Ω and kn > k0,

zkn1 (x) ≤ ε or zkn2 (x) ≤ ε.

Letα = maxi=1,2 supx∈Ω |αi(x)| > 0; then for each x ∈ Ω and kn > k0,

α2zkn1 (x) ≤αε or α1z

kn2 (x) ≤αε.

If wkn(x) ≥ 0, then (wkn)+(x) = wkn(x), (wkn)−(x) = 0 and α2zkn1 (x) ≥ α1z

kn2 (x) ≥ 0, and hence α1z

kn2 (x) ≤ αε. It follows

that

|(wkn)+(x) − α2zkn1 (x)| ≤αε and |(wkn)−(x) + α1z

kn2 (x)| ≤αε.

If wkn(x) ≤ 0, a similar argument gives the same result.(ii) It follows from Theorem 2.3, Lemma 3.1 and (4.1).(iii) Lemmas 3.2, 3.4 and 3.5 yields the result.(iv) Multiplying (4.1) by any φ ∈ W 1,2

0 (Ω) and integrating over Ω gives

−

Ω

∇wkn · ∇φdx =

Ω

zkn1 1α2 − zkn2 1α1 + 2∇α2 · ∇zkn1 − 2∇α1 · ∇zkn2 − α2f1(z

kn1 , zkn2 )

a1 − b1f1(z

kn1 , zkn2 )

+ α1f2(z

kn1 , zkn2 )

a2 − b2f2(z

kn1 , zkn2 )

φdx,

and using (iii) and Lemma 3.2, passing to the limit as kn → ∞, we obtain the limit equation.(v) The fact that w satisfies (4.4) implies that w ∈ W 2,2(Ω). Using the Sobolev imbedding theorem, we have that

w ∈ W 1,p(Ω) for some p > 2; then (4.4) implies that w ∈ W 2,p(Ω). Continuing this process gives that w ∈ W 2,p(Ω)for all p ∈ [1, ∞).

3106 L. Zhou et al. / Nonlinear Analysis 75 (2012) 3099–3106

Remarks. (i) In the proof of Theorem 4.1(iv), the weak convergence results given in Lemma 3.3 are sufficient.(ii) Using Theorem 4.1(i), we have the following uniform convergence result. Up to a subsequence,

zk1 → z1, zk2 → z2 uniformly on Ω.

In fact, if N 6 3, we have that wk is bounded in L2(Ω) uniformly in k, and hence wk is uniformly bounded in W 2,2(Ω)and therefore in a Hölder space (by Sobolev’s embedding theorem). This implies that there exists a subsequence of wk thatconverges uniformly to w on Ω and the result follows by using Theorem 4.1(i). If N > 3, we have that wk is uniformlybounded in W 2,2(Ω). Using Sobolev’s imbedding theorem, we have that w ∈ W 1,p(Ω) for some p > 2, and then thefirst equation in (4.1) implies that w ∈ W 2,p(Ω). Continuing this process gives that w ∈ W 2,p(Ω) for all p ∈ [1, ∞).Using Sobolev’s imbedding theorem again, we then have that wk is uniformly bounded in a Hölder space, and the uniformconvergence result follows from Theorem 4.1(i).

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