MONOTONE ITERATIVE TECHNIQUE FOR SEMILINEAR ELLIPTIC SYSTEMS A. S. VATSALA AND JIE YANG Received 27 pdf

We prove that the monotone sequences converge to coupled weak minimal and maximal solutions of the nonlinear elliptic systems.. Using generalized monotone method, the existence and uniqu

FOR SEMILINEAR ELLIPTIC SYSTEMS

A S VATSALA AND JIE YANG

Received 27 September 2004 and in revised form 23 January 2005

We develop monotone iterative technique for a system of semilinear elliptic boundary value problems when the forcing function is the sum of Caratheodory functions which are nondecreasing and nonincreasing, respectively The splitting of the forcing function leads to four different types of coupled weak upper and lower solutions In this paper, rel-ative to two of these coupled upper and lower solutions, we develop monotone iterrel-ative technique We prove that the monotone sequences converge to coupled weak minimal and maximal solutions of the nonlinear elliptic systems One can develop results for the other two types on the same lines We further prove that the linear iterates of the mono-tone iterative technique converge monotonically to the unique solution of the nonlinear BVP under suitable conditions

1 Introduction

Semilinear systems of elliptic equations arise in a variety of physical contexts, specially in the study of steady-state solutions of time-dependent problems See [1,4,5], for exam-ple Existence and uniqueness of classical solutions of such systems by monotone method has been established in [2,4] Using generalized monotone method, the existence and uniqueness of coupled weak minimal and maximal solutions for the scalar semilinear elliptic equation has been established in [3] They have utilized the existence and unique-ness result of weak solution of the linear equation from [1] In [3], the authors have con-sidered coupled upper and lower solutions and have obtained natural sequences as well

as alternate sequences which converge to coupled weak minimal and maximal solutions

of the scalar semilinear elliptic equation

In this paper, we develop generalized monotone method combined with the method

of upper and lower solutions for the system of semilinear elliptic equations For this pur-pose, we have developed a comparison result for the system of semilinear elliptic equa-tions which yield the result of the scalar comparison theorem of [3] as a special case One can derive analog results for the other two types of coupled weak upper and lower solutions on the same lines We develop two main results related to two different types

of coupled weak upper and lower solutions of the nonlinear semilinear elliptic systems

Copyright©2005 Hindawi Publishing Corporation

Boundary Value Problems 2005:2 (2005) 93–106

DOI: 10.1155/BVP.2005.93

We obtain natural as well as intertwined monotone sequences which converge uniformly

to coupled weak minimal and maximal solutions of the semilinear elliptic system Fur-ther using the comparison theorem for the system, we establish the uniqueness of the weak solutions for the nonlinear semilinear elliptic systems The existence of the solution

of the linear system has been obtained as a byproduct of our main results

2 Preliminaries

In this section, we present some known comparison results, existence and uniqueness results related to scalar semilinear elliptic BVP without proofs See [1,3] for details Consider the semilinear elliptic BVP

ᏸu = F(x, u) inU,

whereU is an open, bounded subset of R m andu : U → Ris unknown,u = u(x) Here

F : U → R is known F ∈ L2(U), F(x, u) is a Caratheodory function, that is, F( ·,u) is

measurable for allu ∈ RandF(x, ·) is continuous a.e.x ∈ U.ᏸ denotes a second-order partial differential operator with the divergence form

ᏸu = −

m

i, j =1



a i j(x)u x i



for given coefficient functions ai j(x), c(x) ∈ L ∞(U) (i =1, 2, , m) We assume the

sym-metry conditiona i j = a ji(i, j =1, , m), c(x) ≥0, and the partial differential operator ᏸ

is uniformly elliptic such that there exists a constantθ > 0 such that

m

i, j =1

a i j(x)ξ i ξ j ≥ θ | ξ |2 (2.3)

for a.e.x ∈ U and all ξ ∈ R m

We recall the following definitions for future use

Definition 2.1 (i) The bilinear form B[ ·,·] associated with the divergence form of the elliptic operatorᏸ defined by (2.2) is

B[u, v] =



U


m

i, j =1

a i j(x)u x i v x j+c(x)uv



foru, v ∈ H1(U), where H1(U) is a Sobolev space W1,2(U).

(ii) We say thatu ∈ H1(U) is a weak solution of the boundary value problem (2.1) if

for allv ∈ H1(U), where ( ·,·) denotes the inner product inL2(U).

Definition 2.2 The function α0∈ H1(U) is said to be a weak lower solution of (2.1) if,

α0≤0 on∂U and



U


m

i, j =1

a i j(x)α0,x i v x j+c(x)α0v



dx ≤



U F

x, α0



for eachv ∈ H1(U), v ≥0 If the inequalities are reversed, thenα0is said to be a weak upper solution of (2.1)

In order to discuss the results on monotone iterative technique, we need to consider the existence and uniqueness of weak solutions of linear boundary value problems The result on the existence of weak solutions for the linear BVP can be obtained from the Lax-Milgram theorem which is stated below In the following theorem, we assume that

H is a real Hilbert space, with norm  · and inner product (·,·), we let·,·denote the pairing ofH with its dual space.

Theorem 2.3 (the Lax-Milgram theorem) Assume that B : H × H → R is a bilinear map-ping, for which there exist constants α, β > 0 such that

(i)| B[u, v] | ≤ α  u  v  , u, v ∈ H;

(ii)β  u 2≤ B[u, u], u ∈ H.

Also assume that F : H → R is a bounded linear functional on H.

Then there exists a unique element u ∈ H such that

for all v ∈ H.

The following theorem proves the unique solution of the linear BVP, which is [3, The-orem 5.2.4]

Theorem 2.4 Consider the linear BVP

ᏸu = h(x) in U,

Then there exists a unique solution u ∈ H1(U) for the linear BVP ( 2.8 ) provided 0 < c ∗ ≤ c(x) a.e in U and h ∈ L2(U).

The next theorem is a comparison theorem, a modified version of which is needed in our main results This is [3, Theorem 5.2.5]

Theorem 2.5 Let α0,β0be weak lower and upper solutions of ( 2.1 ) Suppose further that F satisfies

F

x, u1



− F

x, u2



≤ K

u1− u2



(2.9)

whenever u1≥ u2a.e for x ∈ U and K(x) > 0 for x ∈ U Then, if 0 < c − K ∈ L1(U),

The following corollary is the special case ofTheorem 2.5

Corollary 2.6 For p ∈ H1(U) satisfying



U

 m

i, j =1

a i j(x)p x i v x j+c(x)pv



for each v ∈ H1(U), v ≥ 0 a.e and p ≤ 0 on ∂U, p(x) ≤ 0 in U a.e provided c(x) > 0.

The next two theorems [1] are needed to prove that a bounded sequence in a Hilbert space contains a weakly, uniformly convergent subsequence

Theorem 2.7 (weak compactness) Let X be a reflexive Banach space and suppose that the sequence { u k } ∞

k =1∈ X is bounded Then there exist a subsequence { u k j } ∞

j =1⊆ { u k } ∞

k =1and

u ∈ X such that { u k j } ∞

j =1converges weakly to u ∈ X.

Theorem 2.8 (the Ascoli-Arzela theorem) Suppose that { f k } ∞

k =1 is a sequence of real-valued functions defined onRn such that

f k(x)  ≤ M 

k =1, 2, , x ∈ R n

(2.12)

for some constant M, and the { f k } ∞

k =1are uniformly equicontinuous, then there exist a sub-sequence { f k j } ∞

j =1⊆ { f k } ∞

k =1 and a continuous function f such that f k j → f uniformly on compact subset ofRn

3 Main results

In this section, we develop monotone iterative technique for system of semilinear elliptic BVP The results of [3] will be a special case of our results for the scalar semilinear elliptic BVP

We first consider the following system of semilinear elliptic BVP in the divergence form

ᏸu = f (x, u) + g(x, u) inU,

where u : U → R N, ᏸu =(ᏸ1u1,ᏸ2u2, ,ᏸN u N), and ᏸk u k = −(m

i, j =1a k i j(x)u k

i)x j +

c k(x)u k with the bilinear formB[u k,v k]= U(m

i, j =1a k

i j(x)u k

i v k

j+c k(x)u k v k)dx for k =

1, 2, , N Here f , g : U × R N → R Nare Caratheodory functions Other assumptions on

a k i j,c kare the same as fora i j,c inSection 2

In this paper, here and throughout, we assume all the inequalities to be componentwise unless otherwise stated

In order to develop monotone iterative technique for the BVP (3.1), we need to prove the following comparisonLemma 3.1relative to the elliptic system

ᏸu = F(x, u) inU,

where assumption forᏸu, ᏸ k u k,B[u k,v k] are the same as they are in (3.1)

Lemma 3.1 Let α0,β0 be weak lower and upper solutions of ( 3.2 ) when F : U × R N →

RN,u ∈ H1(U) Suppose further that F(x, u) is quasimonotone nondecreasing in u for each component k and satisfies

F k

x, u1,u2, , u N

− F k

x, v1,v2, , v N

≤ K k

N

i =1



u i − v i

(3.3)

whenever u ≥ v a.e for x ∈ U and K k > 0 for k =1, 2, , N Then, if 0 < c k − NK ∈ L1(U), where K =maxK k for k =1, 2, , N,

α k

0(x) ≤ β k

0(x) in U a.e for k =1, 2, , N. (3.4)

Proof From the definition of weak lower and upper solutions, we get



U

 m

i, j =1

a k i j(x)

α k

0,x i − β k

0,x i



v k

j+c k(x)

α k

0− β k

0



v k



dx ≤



U F k

x, α0



− F k

x, β0



v k dx

(3.5)

for eachv k ∈ H1(U), v k ≥0 a.e and k =1, 2, , N Choose v k =(α k0− β k0)+∈ H1(U),

v k ≥0 a.e

Since



α k

0− β k

0

+

x j =







α k0,x j − β k0,x j a.e onα k0> β k0,

using the ellipticity condition (2.3), and (3.3), we integrate (3.5) on the region where

α k0> β0k, fork =1, 2, , N, and we have



α0>β0



θ kα k

0,x i − β k0,x i 2

+c k(x)α k

0− β k0 2

dx ≤



α0>β0

K k

N

i =1



α i0− β i0

α k0− β k0

dx.

(3.7)

We haveN such inequalities for k =1, 2, , N When we add all N inequalities together,

we obtain



α0>β0


N

k =1

θ kα k

0,x i − β0,k x i 2

+

N

k =1

c k(x)α k

0− β k0 2



dx

≤



α0>β0

N

k =1

α k

0− β k

0 
N

k =1

K k

α k

0− β k

0



dx,



α0>β0


N

k =1

θ kα k

0,x i − β k

0,x i 2 +

N

k =1

c k(x)α k

0− β k

0  2



dx ≤



α0>β0

NK

N

k =1

α k

0− β k

0  2

dx,



α0>β0

N

k =1



θ kα k

0,x i − β k

0,x i 2 +

c k(x) − NKα k

0− β k

0  2

dx ≤0.

(3.8) From our assumption, the integrand is nonnegative Hence, the only possibility to keep our inequalities hold true is that the domain of integration is an empty set Hence, we

If, in (3.2),F(x, u) = A(x)u, where A(x) is an N × N matrix, we have the following

corollary for the linear system

Corollary 3.2 Let F(x, u) = A(x)u in ( 3.2 ) and all the assumptions of Lemma 3.1 hold, further let

A(x)u − A(x)v ≤K1,K2, , K N
N

i =1



u i − v i

(3.9)

whenever u ≥ v a.e for x ∈ U and K k > 0 for k =1, 2, , N Then, if 0 < c k − NK k ∈

L1(U), where K k =max(| a k1 |,| a k2 |, , | a kN | ) for k =1, 2, , N,

α k0(x) ≤ β k0(x) in U, a.e for k =1, 2, , N. (3.10) The next corollary is a special application ofLemma 3.1

Corollary 3.3 For p k ∈ H1(U), k =1, 2, , N, satisfying



U

N

k =1


m

i, j =1

a k i j(x)p k i v k j+c k0(x)p k v k



for each v k ∈ H1(U), v k ≥ 0 a.e and p k ≤ 0 on ∂U, then p k(x) ≤ 0 in U a.e provided that

c k0> 0 for x ∈ U, k =1, 2, , N.

Next, we define two types of coupled weak lower and upper solutions of (3.1) In order

to avoid monotony, our main results are developed relative to these two types of coupled weak lower and upper solutions only

Definition 3.4 Relative to the BVP (3.1), the functionsα0,β0∈ H1(U) are said to be

(i) coupled weak lower and upper solutions of type I if

B α k0,v k

≤f k

x, α0

 +g k

x, β0

 ,v k ,

B β k0,v k

≥f k

x, β0

 +g k

x, α0

 ,v k

for eachv k ∈ H1(U), v k ≥0 a.e inU and k =1, 2, , N;

(ii) coupled weak lower and upper solutions of type II if

B α k

0,v k

≤f k

x, β0

 +g k

x, α0

 ,v k ,

B β k

0,v k

≥f k

x, α0

 +g k

x, β0

 ,v k

for eachv k ∈ H1(U), v k ≥0 a.e inU and k =1, 2, , N.

We are now in a position to prove the first main result on monotone method for the system of elliptic BVP (3.1)

Theorem 3.5 Assume that

(A1)α0,β0∈ H1(U) are the coupled weak lower and upper solutions of type I with α0(x) ≤

β0(x) a.e in U × R N ;

(A2) f , g : U × R N → R N are Caratheodory functions such that f k(x, u) is nondecreasing

in each component u i , g k(x, u) is nonincreasing in each component u i for x ∈ U a.e where i, k =1, 2, , N;

(A3)c k(x) ≥ c ∗ k > 0 in U a.e and for any η, µ ∈ H1(U × R N ) with α0≤ η, µ ≤ β0, the function h k(x) = f k(x, η) + g k(x, µ) ∈ L2(U) for k =1, 2, , N.

Then for any solution u(x) of BVP ( 3.1 ) with α0(x) ≤ u(x) ≤ β0(x), there exist monotone sequences { α n(x) },{ β n(x) } ∈ H1(U × R N ) such that α k

n
ρ k , β k

n
γ k weakly in H1(U) as

n → ∞ and (ρ, γ) are coupled weak minimal and maximal solutions of ( 3.1 ), respectively, that is,

ᏸk ρ k = f k(x, ρ) + g k(x, γ) in U, ρ k = 0 on ∂U,

ᏸk γ k = f k(x, γ) + g k(x, ρ) in U, γ k = 0 on ∂U, (3.14) for k =1, 2, , N.

Note Here and inTheorem 3.8, when we say thatρ, γ are coupled weak solutions means

that they satisfy the following variational form:

B ρ k,v k

=



U f k(x, ρ) + g k(x, γ)

v k dx,

B γ k,v k

=



U f k(x, γ) + g k(x, ρ)

v k dx.

(3.15)

Proof Consider the linear BVP

ᏸk α k n+1 = f k

x, α n

 +g k

x, β n



inU, α k n+1 =0 on∂U,

ᏸk β k n+1 = f k

x, β n

 +g k

x, α n



inU, β k n+1 =0 on∂U, (3.16)

wheren =0, 1, The variational forms associated with (3.16) are

B α k n+1,v k

=



U f k

x, α n +g k

x, β n

v k dx,

B β k n+1,v k

=



U f k

x, β n +g k

x, α n

v k dx,

(3.17)

for allv k ∈ H1(U), v k ≥0 a.e inU for k =1, 2, , N.

We want to show that the weak solutionsα n,β n of (3.16) are uniquely defined and satisfy

α0≤ α1≤ ··· ≤ α n ≤ β n ≤ ··· ≤ β1≤ β0 a.e inU. (3.18) For eachn ≥1, if we haveα0≤ α n ≤ β n ≤ β0, then by hypothesis (A3), h k

1(x) = f k(x, α n) +

g k(x, β n)∈ L2(U), h k

2(x) = f k(x, β n) +g k(x, α n)∈ L2(U), and c k(x) ≥ c k ∗ > 0 Hence,

Theorem 2.4 implies that BVP (3.16) has unique weak solution α k

n and β k

n for k =1,

2, , N.

In order to show that (3.18) is true, we first prove thatα k

1≥ α k

0a.e inU for each kth

component Now letp k = α k0− α k1so that p k ≤0 on∂U and for v k ∈ H1(U), v k ≥0 a.e

inU, by the definition of type I of coupled weak lower and upper solutions, we have

B p k,v k

= B α k0,v k

− B α k1,v k

≤



U f k

x, α0

 +g k

x, β0



v k dx −



U f k

x, α0

 +g k

x, β0



v k dx =0. (3.19)

Hence, byCorollary 2.6,p k ≤0 inU a.e., that is, α k0≤ α k1inU a.e Similarly, we can show

thatβ k1≤ β k0a.e inU, where k =1, 2, , N.

Assume, for some fixedn > 1, α n ≤ α n+1andβ n ≥ β n+1a.e inU Now consider p k =

α k n+1 − α k n+2, withp k =0 on∂U, and using the monotone properties of f , g, we get

B p k,v k

=



U f k

x, α n +g k

x, β n

− f k

x, α n+1

− g k

x, β n+1

v k dx ≤0. (3.20)

ByCorollary 2.6, we getα k n+1 ≤ α k n+2a.e inU Similarly, we can show that β k n+1 ≥ β k n+2a.e

inU componentwise Hence, using the induction argument, we get α k n −1≤ α k

n,β n k −1≥ β k

n

a.e inU for all n ≥1

Now we want to show thatα1≤ β1a.e inU Consider p k = α k1− β k1andp k =0 on∂U.

Sinceα0≤ β0, by the monotone properties of f , g, we have

B p k,v k

=



U f k

x, α0

 +g k

x, β0



− f k

x, β0



− g k

x, α0



v k dx ≤0. (3.21) Hence,α k

1≤ β k

1a.e inU for k =1, 2, , N byCorollary 2.6

Assumeα k

n ≤ β k

na.e inU for some fixed n > 1 We can also prove α k

n+1 ≤ β k n+1a.e inU

using similar argument By induction, (3.18) holds forn ≥1

Since monotone sequences{ α n },{ β n } ∈ H1(U × R N), there exist pointwise limits for each componentk, where k =1, 2, , N That is,

lim

n →∞ α k

n(x) = ρ k(x) a.e inU, lim

n →∞ β k

n(x) = γ k(x) a.e inU, (3.22)

whereρ k,γ k ∈ H1(U), since a Hilbert space is a Banach space which is a complete, normed

linear space

For eachn ≥1, we note that for eachv k ∈ H1(U), α k

nsatisfies



U


m

i, j =1

a k

i j(x)

α k n



x i v x j+c k(x)α k

n v k



dx =



U f k

x, α n −1

 +g k

x, β n −1



v k dx. (3.23)

We now use the ellipticity condition and the fact thatc k(x) ≥ c ∗ k(x) > 0 with v k = α k

nto get



U θ k

α k n,x2

+c k ∗(x)

α k n2

dx ≤



U f k

x, α n −1

 +g k

x, β n −1



v k dx. (3.24) Since the integrand on the right-hand side belongs toL2(U), we obtain the estimate

sup

n

α k

n

Hence, there exists a subsequence{ α k

n i }which converges weakly toρ k(x) in H1(U) by

Theorem 2.7 Similarly, we can show that supn  β k

n  H1 (U) < ∞ Hence, there exists a sub-sequence{ β k

n i }which converges weakly toγ k(x) in H1(U) usingTheorem 2.7

Sequence{ α k

n(x) }mapsU intoRfor eachk =1, 2, , N It is easy through

contradic-tion method to show that for eachε > 0, there exists δ > 0 such that | x − y | < δ implies

that  α k

n(x) − α k

n(x0) W1,2 (U) < ε for x, y ∈ U Hence, { α k

n(x) }is equicontinuous on U.

Similarly, we can show that{ β k

n(x) }is also equicontinuous onU Then by the Ascoli–

Arzela theorem, the subsequences{ α k

n i },{ β k

n i }converge uniformly onU Since both of

the sequences{ α k

n(x) },{ β k

n(x) }are monotone, the entire sequences converge uniformly and weakly toρ k(x), γ k(x), respectively, on U for k =1, 2, , N Therefore, taking the

limit asn → ∞for (3.17), we obtain

B ρ k,v k

=



U f k(x, ρ) + g k(x, γ)

v k dx,

B γ k,v k

=



U f k(x, γ) + g k(x, ρ)]v k dx.

(3.26)

Hence,ρ, γ are the coupled weak solutions of (3.1) Finally, we want to prove thatρ and

γ are the coupled weak minimal and maximal solutions of (3.1) That is, ifu is any weak

solution of (3.1) such thatα0(x) ≤ u(x) ≤ β0(x) a.e in U × R N, then the following claim will be true Fork =1, 2, , N,

α k0(x) ≤ ρ k(x) ≤ u k(x) ≤ γ k(x) ≤ β k0(x) a.e inU. (3.27)

To prove that for any fixed n ≥1,α k

n(x) ≤ u k(x) ≤ β k

n(x) a.e in U, we assume that for

some fixedn ≥1, α k

n(x) ≤ u k(x) ≤ β k

n(x) a.e in U is true, since α0(x) ≤ u(x) ≤ β0(x) is

claimed from the hypothesis Letp k = α k n+1 − u k, withp k =0 on∂U Using the monotone

properties of f , g, we obtain

B p k,v k

=



U f k

x, α n

 +g k

x, β n



− f k(x, u) − g k(x, u)

v k dx ≤0. (3.28)

Hence, byCorollary 2.6,α k

n+1 ≤ u k a.e inU In a similar way, we obtain u k ≤ β k

n+1 By induction,α k

n(x) ≤ u k(x) ≤ β k

n(x) a.e in U for all n ≥1 Now taking the limit ofα k

n,β k

nas

Remark 3.6 (i) When N =1, the results ofTheorem 3.5 yield the scalar result of [3], which is [3, Theorem 5.2.1]

(ii) In (3.1), ifg(x, u) ≡0,f (x, u) is not nondecreasing in some u kcomponents, where

k =1, 2, , N, then we can construct f k(x, u) = f k(x, u) + d k u kwhich is nondecreasing

in eachu kwithd k ≥0 Letg k(x, u) = − d k u kwhich is nonincreasing inu k Then we can solve the BVP

ᏸk u k = −

 m

i, j =1

a k i j(x)u k

i



x j

+c k(x)u k = f k(x, u) + g k(x, u), (3.29)

where (f ) k(x, u) is nondecreasing in each u l,g k(x, u) is nonincreasing in each u lforl, k =

1, 2, , N Assume that the type-I coupled weak upper lower solutions of (3.1) are also the type-I coupled weak upper lower solutions of the new constructed elliptic BVP (3.29), thenTheorem 3.5still can be applied to (3.29) and the solutions of (3.29) will be the solutions for (3.1)

(iii) In (3.1), if f (x, u) ≡0,g(x, u) is not nonincreasing in some u kcomponents, where

k =1, 2, , N, then we can construct g k(x, u) = g k(x, u) − d k u kwhich is nondecreasing in eachu kwithd k ≥0 Let f k(x, u) = d k u kwhich is nondecreasing inu k Then we can solve the BVP

ᏸk u k = −


m

i, j =1

a k i j(x)u k

i



x j

+c k(x)u k = f k(x, u) + g k(x, u), (3.30)

where f k(x, u) is nondecreasing in each u l,g k(x, u) is nonincreasing in each u lforl, k =

1, 2, , N Assume that the type I coupled with upper lower solutions of (3.1) are also the type-I coupled weak upper lower solutions of the new constructed elliptic BVP (3.30), then applyTheorem 3.5to (3.30) and get the solutions we need for (3.1)

(iv) Other varieties on the properties of f (x, u), g(x, u) such as f (x, u) is not

nonde-ceasing in everyu k component andg(x, u) is not nonincreasing in every u kcomponent,

we can always use the idea in (ii), (iii) to solve the new constructed elliptic BVP

un-der suitable assumption of coupled upper and lower solutions for the newly constructed problem

The following corollary is to show the uniqueness of the solution for (3.1)

Corollary 3.7 Assume, in addition to the conditions of Theorem 3.5 , f and g satisfy

f k

x, u1,u2, , u N

− f k

x, v1,v2, , v N

≤ N1

N

i =1



u i − v i ,

g k

x, u1,u2, , u N

− g k

x, v1,v2, , v N

≥ − N2

N

i =1



u i − v i ,

(3.31)