# Positive Solutions for Some Nonlinear Elliptic Systems in Exterior

Authors: Ramzi Alsaedi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Using some potential theory tools and the Schauder fixed point theorem, we prove
the existence of positive continuous solutions with a precise global behavior
for the competitive semilinear elliptic systemΔu=p(x)uαvr, Δv=q(x)usvβ in an exterior domain D of ℝ2, subject to some Dirichlet conditions, where
α≥1, β≥1, r≥0, s≥0 and the potentials p,q are nonnegative and satisfy some hypotheses
related to the Kato class K(D).


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## Body

## 1. Introduction

The study of nonlinear elliptic systems has a strong motivation, and important
research efforts have been made recently for these systems aiming to apply the
results of existence and asymptotic behavior of positive solutions in applied
fields. Coupled nonlinear Shrödinger systems arise in the description of
several physical phenomena such as the propagation of pulses in birefringent optical
fibers and Kerr-like photorefractive media, see [1, 2]. Stationary elliptic
systems arise also in other physical models like non-Newtonian fluids: pseudoplastic
fluids and dilatant fluids [3, 4], non-Newtonian filtration [5], and the turbulent flow of a gas in porous
medium [6, 7]. They also describe other various nonlinear phenomena such as
chemical reactions, pattern formation, population evolution where, for example,
u and v represent the concentrations of two species in the
process. As a consequence, positive solutions of such are of interest.For some recent results on the qualitative analysis and the applications of positive
solutions of nonlinear elliptic systems in both bounded and unbounded domains we
refer to [8–15] and the references therein.In these works various existence results of positive bounded solutions or positive
blowing-up ones (called also large solutions) have been established, and a precise
global behavior is given. We note also that several methods have been used to treat
these nonlinear systems such as sub- and super-solutions method, variational method,
and topological methods.In this paper, we consider an unbounded domainD in ℝ2 with a nonempty compact boundary
∂D consisting of finitely many Jordan curves and
noncontaining zero. We fix two nontrivial nonnegative continuous functions
φ and ψ on ∂D and some nonnegative constants,
a, b, c, d such that a+c>0 and b+d>0, and we will deal with the existence of a positive
continuous solution (in the sense of distributions) to the system:(1.1)Δu=p(x)uαvr,inDΔv=q(x)usvβ,inDu/∂D=aφ,v/∂D=bψlim⁡|x|→∞u(x)ln⁡|x|=c,lim⁡|x|→∞v(x)ln⁡|x|=d, where α≥1, β≥1, r≥0, s≥0 and p,q are two nonnegative functions satisfying some
hypotheses related to the Kato class K(D) defined and studied in [16, 17] by means of the
Green function G(x,y) of the Dirichlet Laplacian in
D.Our method is based on some potential theory tools which we apply to give an
existence result for equations by an approximation argument, then we use the result
for equations to prove, by means of the Schauder fixed point theorem, the existence
result for the system (1.1).As far as we know, there are no results that contain existence of positive solutions
to the elliptic system (1.1) in the
case where α>0 and β>0 and the weights p(x) and q(x) are singular functions.The study of (1.1) is motivated by
the existence results obtained in [18] to the
following system(1.2)Δu=λp(x)g(v),inDΔv=μq(x)f(u),inDu/∂D=aφ,v/∂D=bψ,lim⁡|x|→∞u(x)ln⁡(|x|)=c,lim⁡|x|→∞v(x)ln⁡(|x|)=d, where λ,μ are nonnegative constants, the functions
f,g:[0,∞)→[0,∞) are nondecreasing and continuous.More precisely, it was shown in [18] that if
the functions p̃:=pf(θ) and q̃:=qg(ω) belong to the Kato class K(D), then there exist λ0>0 and μ0>0 such that for each λ∈[0,λ0) and μ∈[0,μ0), the system (1.2) has a positive continuous solution
(u,v) having the global asymptotic behavior of the unique
solution of the associated homogeneous system, where we have denoted by(1.3)ω:=aHDφ+ch,θ:=bHDψ+dh, and the functions HDφ and h are the harmonic functions defined, respectively,
by (1.4) and (1.5) below.The system of two equations in (1.1)
has been treated in exterior domains of ℝn, n≥3 in [19], and
existence of positive bounded continuous solutions is established. The main
difficulty in the present work is the case of the domain D⊂ℝ2. More precisely, the function
h defined by (1.5) behaves as ln(|x|) at infinity for n=2, unlike the case n≥3 where this function is bounded at infinity.Throughout this paper, we denote byHDφ the unique bounded continuous solution of the
Dirichlet problem(1.4)Δw=0inD,w/∂D=φ,lim⁡|x|→+∞w(x)h(x)=0, where φ is a nonnegative continuous function on
∂D.The functionh is defined on D by(1.5)h(x)=2πlim⁡|y|→+∞GD(x,y). First, we recall the following result about this
function h.Proposition 1.1 (see [16]).
The functionh defined by (1.5) is harmonic and positive in D and satisfies (1.6)lim⁡x→z∈∂Dh(x)=0,lim⁡|x|→+∞h(x)ln⁡|x|=1.Taking into account these notations, we use some potential theory tools and an
approximating sequence in order to prove the following first result concerning the
existence of a unique positive continuous solution to the boundary value
problem:(1.7)Δu=p(x)uγ,inDu/∂D=aφlim⁡x→∞u(x)ln⁡|x|=c, where γ≥1, φ is a nontrivial nonnegative continuous function on
∂D and a,c are two nonnegative constants with
a+c>0. More precisely we establish the following.Theorem 1.2.
Letp be a nonnegative function such that the
function p̃=γpωγ-1 belongs to the Kato class
K(D). Then problem (1.7) has a unique positive continuous solution satisfying
for each x∈D(1.8)c0ω(x)≤u(x)≤ω(x), where ω is defined in (1.3) and the constant c0∈(0,1].Next we exploit this result to prove the existence of a positive continuous solution(u,v) to the system (1.1). For this aim we denote by(1.9)ω0:=a+ch,(1.10)θ0:=b+dh, and we need to assume the following hypothesis on the
functions p and q.(H)
p and q are nonnegative measurable functions in
D such that (1.11)x⟶p(x)θ0r(x)ω0α-1(x),x⟶q(x)ω0s(x)θ0β-1(x) are in K(D).Using the Schauder fixed point, we prove the following main result.Theorem 1.3.
Under the hypothesis(H), the problem (1.1) has a positive continuous solution
(u,v) satisfying for each x in D(1.12)c1ω(x)≤u(x)≤ω(x),c2θ(x)≤v(x)≤θ(x), where ω, θ are defined by (1.3) and c1,c2∈(0,1].In order to state these results and for the sake of completeness, we give in the
sequel some notations, and we recall some properties of the Kato classK(D) studied in [16, 17].Let us denote byB(D) the set of Borel measurable functions in
D and by B+(D) the set of nonnegative ones. We denote also by
C0(D) the set of continuous functions in
D having limit zero at ∂D, by C(D¯∪{∞})={f∈C(D¯):lim|x|→∞f(x)exists} and by C0(D¯)={f∈C(D¯):lim|x|→∞f(x)=0}. We note that C(D¯∪{∞}) is a Banach space endowed with the uniform norm
∥f∥∞=supx∈D|f(x)|.First we recall that ifφ is a nonnegative continuous function on
∂D, then from [20, page 427] the function HDφ∈C(D¯∪{∞}) and satisfies limx→∞HDφ(x)=C>0.For anyf in B+(D), we denote by Vf the Green potential of f defined on D by (1.13)Vf(x):=∫DG(x,y)f(y)dy, and we recall that if f∈Lloc1(D) and Vf∈Lloc1(D), then we have in the distributional sense (see
[21, page 52]) (1.14)Δ(Vf)=-finD. Furthermore, we recall that for
f∈B+(D), the potential Vf is lower semicontinuous in
D and if f=f1+f2 with f1,f2∈B+(D) and Vf∈C+(D), then Vfi∈C+(D) for i∈{1,2}.Let(Xt,t>0) be the Brownian motion in ℝ2 and Px be a probability measure on the Brownian continuous
paths starting at x. For any function q∈B+(D), we define the kernel Vq by(1.15)Vqf(x)=Ex(∫0τDe-∫0tq(Xs)dsf(Xt)dt), where Ex is the expectation on Px and τD=inf{t>0:Xt∉D}.Ifq is a nonnegative function in
D such that Vq<∞, the kernel Vq satisfies the following resolvent equation (see
[21, 22])(1.16)V=Vq+Vq(qV)=Vq+V(qVq). So for each u∈B(D) such that V(q|u|)<∞, we have(1.17)(I-Vq(q.))(I+V(q.))u=(I+V(q.))(I-Vq(q.))u=u, and for each u∈B+(D), we have(1.18)0≤Vq(u)≤V(u). Now we recall the definition of the Kato class which
contains in particular a wider class of singular functions near the boundary of the
domain D.Definition 1.4 (see [16]).
A Borel measurable functions in D belongs to the Kato class
K(D) if (1.19)lim⁡α→0sup⁡x∈D∫D∩B(x,α)ρ(y)ρ(x)G(x,y)|s(y)|dy=0,lim⁡M→∞sup⁡x∈D∫D∩{|y|≥M}ρ(y)ρ(x)G(x,y)|s(y)|dy=0, where ρ(x)=min(1,δ(x)) and δ(x) denotes the Euclidian distance from
x to the boundary ∂D of D.This Kato class is rich enough as it can be seen in the following example.Example 1.5 (see [16]).
Letq(x)=1/(1+|x|)μ-λ(δ(x))λforx∈D.Then (1.20)q∈K(D)ifandonlyifλ<2<μ⋅Remark 1.6.
Letp>1 and λ,μ∈ℝ such that λ<2-(2/p)<μ. Then using the Hölder inequality and the
same arguments as in the proof of the precedent example it follows that for each
f∈Lp(D), the function defined in
D by f(x)/(1+|x|)μ-λ(δ(x))λ belongs to K(D).Next, we recall some properties ofK(D).Proposition 1.7 (see [16, 17]).
Letq be a nonnegative function in
K(D). Then one has(i)
αq:=supx,y∈D∫D(G(x,z)G(z,y)/G(x,y))q(z)dz<∞.(ii)
The functionx→(δ(x)/(1+|x|))q(x) is in L1(D). In particular
q∈Lloc1(D).(iii)
Vq∈C0(D).(iv)
For any nonnegative superharmonic functionv in D and all x∈D, one has(1.21)∫DG(x,y)v(y)q(y)dy≤αqv(x)⋅The following compactness results will be used and they are proved, respectively, in
[17] and [16].Proposition 1.8 (see [17, Lemma 3.1]).
Leth0 be a positive harmonic function in
D, which is continuous and bounded in
D¯ and let q be a nonnegative function belonging to
K(D). Then the family of functions: (1.22)Fq={∫DGD(⋅,y)h0(y)p(y)dy:|p|≤q} is uniformly bounded and equicontinuous on
D¯∪{∞}. Consequently, it is relatively compact in
C(D¯∪{∞}).Proposition 1.9 (see [16, Lemma 4.3]).
Leth be the function defined by (1.5) and let
q be a nonnegative function in
K(D). Then the family of functions: (1.23)Fq={1h(⋅)∫DG(⋅,y)h(y)p(y)dy;|p|≤q} is uniformly bounded and equicontinuous on
D¯∪{∞}, and consequently it is relatively compact in
C0(D¯).As a consequence of these Propositions, we obtain the following.Corollary 1.10.
Letq be a nonnegative function in
K(D). Then the family of functions: (1.24){x⟶1ω0(x)∫DG(x,y)ω0(y)p(y)dy;|p|≤q} is relatively compact in
C(D¯∪{∞}).Proof.
Since(1.25)ω0(y)ω0(x)=a+ch(y)a+ch(x)≤max⁡(1,h(y)h(x))≤1+h(y)h(x), then the result follows from Propositions 1.8 and 1.9.The following result will play an important role in the proofs of Theorems1.2 and 1.3.Proposition 1.11 (see [17, Proposition
2.9]).
Letv be a nonnegative superharmonic function in
D and q be a nonnegative function in
K(D). Then for each x∈D such that 0<v(x)<∞, one has (1.26)exp⁡(-αq)v(x)≤v(x)-Vq(qv)(x)≤v(x).


## 2. Proof of Theorem1.2

First we give two Lemmas that will be used for uniqueness.Lemma 2.1 (see [23]).
Letζ be a function in B+(D) and ϑ be a nonnegative superharmonic function in
D. Then for all z∈B(D) such that V(ζ|z|)<∞ and z+V(ζz)=ϑ, one has 0≤z≤ϑ.Lemma 2.2.
Letu be a nonnegative continuous function in
D¯∪{∞}. Then (2.1)uisasolutionof(1.7)ifandonlyifu=ω-V(puγ)inD.Proof.
Letu be a nonnegative continuous solution of (1.7). First, we will prove that
u(x)≤ω(x) in D¯. Since HDφ is bounded, then limx→∞((u(x)-aHDφ(x))/h(x))=limx→∞((u(x)-aHDφ(x))/ln|x|)=c. Consequently, for ɛ>0 there exists M>0 such that (2.2)u(x)-aHDφ(x)≤(c+ɛ)h(x)for|x|≥M. This implies that the function
vɛ=ω+ɛh-u satisfies (2.3)Δvɛ=-p(x)uγ≤0inDvɛ=0on⁡∂Dliminf⁡x→∞vɛ(x)≥0. Hence by [20, page 465], we get u(x)≤ω(x)+ɛh(x) in D¯. Since ɛ is arbitrary, this implies that
u(x)≤ω(x) for each x∈D. Now, since p̃=γpωγ-1∈K(D), then puγ-1∈K(D). Hence it follows from Propositions 1.8 and 1.9 that V(aHDpuγ-1) and V(chpuγ-1) belong to C(D¯) with boundary value zero, which implies that
V(pωuγ-1) belongs to C(D¯) with boundary value zero. So,
V(puγ) belongs to C(D¯) with boundary value zero. Consequently, using
Corollary 7 page 294 in [20], we deduce
that the function u-ω+V(puγ) is a classical harmonic in
D with boundary value zero and satisfying
lim|x|→∞((u(x)-ω(x)+V(puγ)(x))/ln|x|)=0. Thus by [20, page 419], we have u-ω+V(puγ)=0 in D. So u=ω-V(puγ) and this proves necessity.
Now, we prove sufficiency. Letu be a nonnegative continuous function in
D¯ satisfying the integral equation
u=ω-V(puγ). Since p is nonnegative and p̃=γpωγ-1∈K(D), then u≤ω and puγ-1∈K(D). This implies, by using Propositions 1.8 and 1.9 that V(aHDφpuγ-1) and V(chpuγ-1) are in C(D¯) with boundary value zero. Consequently,
V(puγ) is in C(D¯) with boundary value zero. Hence,
Δu=Δω-Δ(V(puγ))=puγ (in the sense of distributions) and
u is a solution of (1.7).Now we prove Theorem1.2Proof of Theorem1.2.
First we show that problem (1.7)
has at most one continuous solution. Let u,v be two continuous solutions of (1.7). Then, by Lemma 2.2 we have
u=ω-V(puγ) and v=ω-V(pvγ) in D. Put z=v-u and ζ(x)=(vγ(x)-uγ(x))/(v(x)-u(x)) if u(x)≠v(x) and ζ(x)=0 whenever u(x)=v(x). Then we have ζ≥0 and z+V(pζz)=0 in D. Using Lemma 2.1, we deduce that z=0 and so u=v.
Next, we prove the existence of a positive continuous solution to (1.7). We recall that
ω=aHDφ+ch and p̃=γpωγ-1∈K(D). Put c0=e-αp̃ where the constant αp̃ is defined in Proposition 1.7. We define the nonempty closed
bounded convex set Λ by (2.4)Λ={u∈B+(D):c0ω≤u≤ω}. Let T be the operator defined on
Λ by (2.5)Tu:=ω-Vp̃(p̃ω)+Vp̃(p̃u-puγ). We will prove that T maps Λ to itself. Indeed, for each
u∈Λ, we have (2.6)Tu=ω-Vp̃(p̃ω)+Vp̃(p̃u-puγ)≤ω-Vp̃(puγ)≤ω. On the other hand, since the function
p̃u-puγ≥0, we deduce by Proposition 1.11 that
Tu≥ω-Vp̃(p̃ω)≥c0ω. Hence, TΛ⊂Λ.
Next, we prove thatT is nondecreasing on Λ. Let u1,u2∈Λ such that u1≤u2. Since for each y∈D, the function t→p̃(y)t-p(y)tγ is nondecreasing on [0,ω(y)] we deduce that (2.7)Tu2-Tu1=Vp̃(p̃u2-pu2γ)-Vp̃(p̃u1-pu1γ)=Vp̃[(p̃u2-pu2γ)-(p̃u1-pu1γ)]≥0. Now, we consider the sequence
(uk)k defined by u0=ω-Vp̃(p̃ω) and uk+1=Tuk. Clearly u0∈Λ and u1=Tu0≥u0. Thus, using the fact that
Λ is invariant under T and the monotonicity of
T, we deduce that (2.8)c0ω≤u0≤u1≤⋯≤uk≤ω. Hence, the sequence (uk)k converges to a measurable function
u∈Λ. Therefore, by applying the monotone
convergence theorem, we deduce that u satisfies the following equation: (2.9)u=ω-Vp̃(p̃ω)+Vp̃(p̃u-puγ) or equivalently (2.10)u-Vp̃(p̃u)=ω-Vp̃(p̃ω)-Vp̃(puγ). Applying the operator (I+V(p̃.)) on both sides of (2.10), we deduce by using (1.16) and (1.17) that (2.11)u=ω-V(puγ). Now, let us verify that u is a solution of the problem (1.7). Since
p̃=γpωγ-1∈K(D), then by Proposition 1.7, we have p̃∈Lloc1(D).
Now, using the following inequality:(2.12)puγ≤pωγ≤p̃ω and the continuity of ω in D, we obtain that puγ∈Lloc1(D). Using Proposition 1.7 and (2.12), we obtain for each x∈D(2.13)V(puγ)(x)≤∫DG(x,y)p̃(y)ω(y)dy≤αp̃ω(x), which gives V(puγ)∈Lloc1(D). Thus, by applying Δ on both sides of (2.11), we deduce that u is a solution of (2.14)Δu=puγ(inthesenseofdistributions). Using (2.12), we obtain that puγ and p̃ω are in B+(D). So V(puγ) and V(p̃ω-puγ) are two lower semicontinuous functions. On the
other hand, by Proposition 1.8
we have V(p̃aHDφ)∈C(D) and by Proposition 1.9 the function (1/h)V(p̃h)∈C0(D¯). So V(p̃ω)∈C(D). Thus V(p̃ω-puγ)=V(p̃ω)-V(puγ) is also an upper semicontinuous function.
Consequently, V(p̃ω-puγ)∈C(D). Thus V(puγ)=V(p̃ω)-V(p̃ω-puγ)∈C(D). Therefore u∈C(D). Now, using Propositions 1.1 and 1.9, we deduce that limx→∂DV(p̃h)(x)=0. In addition, since HDφ is bounded in D¯, we deduce from Proposition 1.7  limx→∂DV(p̃)(x)=0. So that limx→∂DV(p̃ω)(x)=0. This in turn implies that
limx→∂DV(puγ)=0. Which together with (2.11) imply that
u/∂D=aφ. On the other hand, we have (2.15)1hV(puγ)≤a‖φ‖∞1hV(p̃)+chV(p̃h). Using Propositions 1.9, 1.8, and 1.1, we
deduce that (1/ln|x|)V(puγ)(x) tends to zero as |x|→∞ and consequently lim|x|→∞(u(x)/ln|x|)=c. This implies that u is a positive continuous solution of (1.7). This completes the proof of
Theorem 1.2.Remark 2.3.
Letp0̃=γmax(1,∥φ∥∞)pω0γ-1, where ω0 is given by (1.9). Then we have 0≤p̃≤p0̃. So if we assume that p0̃∈K(D), then p̃∈K(D) and αp̃≤αp0̃. Moreover, the solution
u of (1.7) satisfies also the inequality: (2.16)e-αp0̃ω≤u≤ωinD.Next we give the proof of Theorem1.3.


## 3. Proof of Theorem1.3

We recall thatω0=a+ch, θ0=b+dhω=aHDφ+ch and θ=bHDψ+dh. Define m=max(1,∥φ∥∞,∥ψ∥∞), f̃=αmr+α-1pθ0rω0α-1 and g̃=βms+β-1qω0sθ0β-1. Put c1=e-αf̃ and c2=e-αg̃, where the nonnegative constants
αf̃ and αg̃ are defined in Proposition 1.7.In order to use a fixed point theorem, we consider the nonempty closed convex setΓ defined by(3.1)Γ={(λ,χ)∈(C(D¯∪{∞}))2:0≤λ≤(1-c1)ωω0,0≤χ≤(1-c2)θθ0}. For (λ,χ)∈Γ, we consider the following system:(3.2)Δy=p(θ-θ0χ)ryα,inDΔz=q(x)(ω-ω0λ)szβ,inDy/∂D=aφ,z/∂D=bψ,lim⁡x→∞y(x)ln⁡|x|=c,lim⁡x→∞z(x)ln⁡|x|=d. Then by Theorem 1.2, the system 21 has a unique positive continuous solution
(y,z) satisfying the integral equations(3.3)y(x)=ω(x)-V(p(θ-θ0χ)ryα)(x),(3.4)z(x)=θ(x)-V(q(ω-ω0λ)szβ)(x). Moreover, we have the following global
inequalities:(3.5)c1ω≤y≤ω,c2θ≤z≤θ,inD,(3.6)p(θ-θ0χ)ryα≤pωαθr≤mαω0f̃,(3.7)q(ω-ω0λ)szβ≤qθβωs≤mβθ0g̃. Let T be the operator defined on
Γ by T(λ,χ)=((ω-y)/ω0,(θ-z)/θ0).Sincey,z satisfy (3.3) and (3.4) we
deduce from (3.5), (3.6), hypothesis
(H) and Corollary 1.10 that the family of functions:(3.8)TΓ={(1ω0V(pyα(θ-θ0χ)r),1θ0V(qzβ(ω-ω0λ)s));(λ,χ)∈Γ} is relatively compact in (C(D¯∪{∞}))2. This together with (3.5) imply that TΓ⊂Γ.Next, we will prove the continuity ofT with respect to the norm ∥·∥ defined on Γ by ∥(λ,χ)∥=∥λ∥∞+∥χ∥∞. Let (λk,χk)k be a sequence in Γ that converges to (λ,χ)∈Γ with respect to ∥·∥. Put (yk,zk)=T(λk,χk) and (y,z)=T(λ,χ). Then we have(3.9)‖(yk,zk)-(y,z)‖=‖yk-yω0‖∞+‖zk-zω0‖∞. Using (3.3), we obtain(3.10)yk-y=V(pyα(θ-θ0χ)r)-V(pykα(θ-θ0χk)r)=V(p[yα((θ-θ0χ)r-(θ-θ0χk)r))+V(p(yα-ykα)(θ-θ0χk)r). Now using the fact that(3.11)tα-ηα=(t-η)[α∫01(η+ξ(t-η))α-1dξ]fort≥0,η≥0, we deduce that(3.12)(yk-y)+V(pk(yk-y))=V(pyα((θ-θ0χ)r-(θ-θ0χk)r)), where pk(x)=αp(x)(θ(x)-θ0(x)χk(x))r∫01[ξyk(x)+(1-ξ)y(x)]α-1dξ.Clearly, we have(3.13)pk(x)≤αp(x)(θ(x)-θ0(x)χk(x))rωα-1(x)≤αp(x)θr(x)ωα-1(x)≤mr+α-1αp(x)θ0r(x)ω0α-1(x)=mr+α-1f̃. Now, since f̃∈K(D) then pk∈K(D) and Vpk<∞. Moreover, we obtain from (3.6) and Proposition 1.7(iv) that(3.14)V(pk|yk-y|)≤V(pykα(θ-θ0χk)r)+V(pyα(θ-θ0χk)r)≤2mαω0αf̃. So we can apply (I-Vpk(pk.)) to (3.12) to obtain from (1.16) and (1.17)
that(3.15)yk-y=Vpk(pyα((θ-θ0χ)r-(θ-θ0χk)r)). On the other hand, we have(3.16)pyα|(θ-θ0χ)r-(θ-θ0χk)r|≤pyα((θ-θ0χ)r+(θ-θ0χk)r)≤2pyαθr≤2mαω0f̃. So from hypothesis (H), Proposition 1.7(iv) and the dominated convergence theorem, we deduce
that for each x∈D, we have(3.17)lim⁡k→∞V(pyα((θ-θ0χ)r-(θ-θ0χk)r))(x)=0. This together with (1.18) and (3.15) implies that for each x∈D, (yk(x))k converges to y(x) as k→∞. Similarly we prove that for each
x∈D, (zk(x))k converges to z(x) as k→∞. Consequently, as TΓ is relatively compact in C(D¯∪{∞}), we deduce that the pointwise convergence implies
the uniform convergence. Namely, ∥(yk-y)/ω0∥∞+∥(zk-z)/θ0∥∞ converges to 0 as k→∞.From the Schauder fixed point theorem there exists(λ,χ)∈Γ such that T(λ,χ)=(λ,χ) or equivalently ((ω-y)/ω0,(θ-z)/θ0)=(λ,χ). Put u=ω-ω0λ and v=θ-θ0χ. Then (u,v) is a positive continuous solution of the system
(1.1) in the sense of
distributions satisfying for each x∈D(3.18)c1ω(x)≤u(x)≤ω(x),c2θ(x)≤v(x)≤θ(x). This completes the proof.



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*Source: 273017-2012-06-04.xml*