The existence of weak solutions for these systems is proved using the theory of monotone operators 1

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS ON R^N

EADA A. EL-ZAHRANI, HASSAN M. SERAG

Abstract. In this paper, we consider the nonlinear elliptic system

−∆_pu=a(x)|u|^p−2u−b(x)|u|^α|v|^βv+f,

−∆qv=−c(x)|u|^α|v|^βu+d(x)|v|^q−2v+g, lim

|x|→∞u= lim

|x|→∞v= 0 u, v >0

on a bounded and unbounded domains of R^N, where ∆p denotes the p- Laplacian. The existence of weak solutions for these systems is proved using the theory of monotone operators

1. Introduction

The generalized (the so-called weak) formulation of many stationary boundary- value problems for partial differential equations leads to operator equation of type

A(u) =f

on a Banach space. Indeed, the weak formulation consists in looking for an unknown functionufrom a Banach spaceV such that an integral identity containinguholds for each test function v from the space V. Since the identity is linear in v, we can take its sides as values of continuous linear functionals at the elementv ∈V. Denoting the terms containing unknownuas the value of an operatorA, we obtain

(A(u), v) = (f, v) ∀v∈V,

which is equivalent to equality of functionals onV, i.e. the equality of elements of V⁰ (the dual space of V): A(u) =f . Functional analysis yields tools for proving existence of generalized (weak) solutions to a relatively wide class of differential equations that appear in mathematical physics and industry.

In our work, we consider nonlinear systems with modelAof the form

A{u, v}={−∆pu−a(x)|u|^p−2u+b(x)|u|^α|v|^βv,−∆qv+c(x)|u|^α|v|^βu−d(x)|v|^q−2v}

These nonlinear systems involving p-Laplacian appear in many problems in pure and applied mathematics e.g. in quasiconformal mappings, non-Newtonian fluids, and nonlinear elasticity [3, 4, 9].

2000Mathematics Subject Classification. 35B45, 35J55.

Key words and phrases. Weak solutions; nonlinear elliptic systems; p-Laplacian;

monotone operators.

c

2006 Texas State University - San Marcos.

Submitted February 9, 2006. Published July 6, 2006.

1

The existence of solutions for such systems was proved, using the method of sub and super solutions in [7, 8, 20]. Here, we use another technique for proving the existence of weak solutions. We use the theory of monotone operators.

First, we consider the following system defined on a bounded domain Ω of R^N with boundary∂Ω:

−∆pu=a(x)|u|^p−2u−b(x)|u|^α|v|^βv+f(x), in Ω

−∆qv=d(x)|v|^q−2v−c(x)|v|^β|u|^αu+g(x), in Ω u=v= 0, on∂Ω.

Then, we generalize the discussion to system defined on the whole spaceR^N. This article is organized as follows: Some technical results and definitions are introduced in section two concerning the theory of nonlinear monotone operators, also, the scalar case is discussed. Section three, is devoted to study the existence of solutions for nonlinear systems defined on a bounded domain. In section four, the existence of solutions for nonlinear systems defined on unbounded domain is proved.

2. Scalar case First, we introduce some technical results [6, 8, 21].

Definitions. LetA:V →V⁰ be an operator on a Banach spaceV. We say that the operatorAis:

Coercive if lim_kuk→∞^hA(u),ui_kuk =∞;

Monotone ifhA(u1)−A(u2), u1−u2i ≥0 for allu1, u2; Strongly continuous ifun

→w uimpliesA(un)→A(u);

Weakly continuous ifu_n →^w uimpliesA(u_n)→^w A(u);

Demicontinuous ifun →uimpliesA(un)→^w A(u).

The operator A is said to satisfy the Mo-condition if un

→w u, A(un) →^w f, and [hA(un), uni → hf, ui] implyA(u) =f.

Theorem 2.1. Let V be a separable reflexive Banach space and A :V → V⁰ an operator which is: coercive, bounded, demicontinuous, and satisfyingMocondition.

Then the equationA(u) =f admits a solution for eachf ∈V⁰.

Now, we prove the existence of a weak solutionu∈W₀^1,p(Ω) for the scalar case

−∆pu=m(x)|u|^p−2u+f(x), x∈Ω,

u= 0 on∂Ω (2.1)

where 0 < a(x) ∈ L^∞(Ω) and Ω is a bounded domain of R^N. In this case, the operatorAisAu=−∆pu−m(x)|u|^p−2u.

It is proved in [2], that if m(x) is a positive function in L^∞(Ω), then the first eigenvalueλ_p(m) of the Dirichlet p-Laplacian problem

−∆_pu=λm(x)|u|^p−2u in Ω

u(x) = 0 on∂Ω (2.2)

is simple, isolated and it is the unique positive eigenvalue having a nonnegative eigenfunction. Moreover it is characterized by

Z

Ω

|∇u|^p≥λp(m) Z

Ω

m(x)|u|^p (2.3)

We prove that (2.1) admits a weak solution ifλp(m)>1 . First, we prove that A is a bounded operator:

(Au, v) = Z

Ω

|∇u|^p−2∇u∇v− Z

Ω

m(x)|u|^p−2uv

Using H¨older’s inequality, we obtain

|(Au, v)| ≤Z

Ω

|∇u|^pp−1_p Z

Ω

|∇v|^p1_p +Z

Ω

m(x)|u|^pp−1_p Z

Ω

m(x)|v|^p_p¹

≤ kuk^p−1_1,p kvk_1,p

To prove that A is continuous, let us assume that un → u in W₀^1,p(Ω). Then kun−uk1,p→0 So that

k∇un− ∇ukp→0 Applying Dominated convergence theorem, we obtain

k|∇un|^p−2∇un− |∇u|^p−2∇ukp →0 hence

kAun−Aukp≤ k|∇un|^p−2∇un− |∇u|^p−2∇ukp+k|un|^p−2un− |u|^p−2ukp→0 OperatorA is strictly monotone:

(Au₁−Au₂, u₁−u₂) = Z

Ω

|∇u1|^p−2∇u1∇u1+ Z

Ω

|∇u2|^p−2∇u2∇u2

− Z

Ω

|∇u1|^p−2∇u1∇u2− Z

Ω

|∇u2|^p−2∇u2∇u1

≥ Z

Ω

|∇u₁|^p+ Z

Ω

|∇u₂|^p−Z

Ω

|∇u₁|^pp−1/pZ

Ω

|∇u₂|^p1/p

−Z

Ω

|∇u₂|^pp−1/pZ

Ω

|∇u₁|^p1/p

=ku1k^p_p+ku2k^p_p− ku1k^p−1_p ku2kp− ku2k^p−1_p ku1kp

= ku1k^p−1_1,p − ku2k^p−1_1,p

ku1k1,p− ku2k1,p

>0. Also,Ais a coercive operator, since from (2.3), we have

(Au, u) = Z

Ω

|∇u|^p− Z

Ω

m|u|^p

≥ Z

Ω

|∇u|^p− 1 λp(m)

Z

Ω

|∇u|^p

=

1− 1 λ_p(m)

Z

Ω

|∇u|^p. Then

(Au, u) kukp

=kuk^p−1_1,p → ∞ as kuk_1,p→ ∞ which proves the existence of a weak solution for (2.1).

3. Nonlinear systems on bounded domains In this section, we consider the system

−∆pu=a(x)|u|^p−2u−b(x)|u|^α|v|^βv+f(x), in Ω

−∆_qv=d(x)|v|^q−2v−c(x)|v|^β|u|^αu+g(x), in Ω u=v= 0, on∂Ω

(3.1)

where Ω is a bounded domain ofR^N, ¹_p+_p¹0 = 1, ¹_q +_q¹0 = 1,α+β+ 2< N and a(x), b(x), c(x), d(x) are positive functions inL^∞(Ω).

Theorem 3.1. For(f, g)∈L^p(Ω)×L^q(Ω), there exists a weak solution (u, v)∈ W₀^1,p(Ω)×W₀^1,q(Ω)for system (3.1)if the following condition is satisfied:

λp(a)>1, and λq(d)>1. (3.2) Proof. We transform the weak formulation of the system (3.1) to the operator form

A(u, v)−B(u, v) =F

where,A, B andF are operators defined onW₀^1,p(Ω)×W₀^1,q(Ω) by (A(u, v),(Φ1,Φ2)) =

Z

Ω

|∇u|^p−2∇u∇Φ1+ Z

Ω

|∇v|^q−2∇v∇Φ2,

(B(u, v),(Φ1,Φ2)) = Z

Ω

a(x)|u|^p−2uΦ1+ Z

Ω

d(x)|v|^q−2vΦ2

− Z

Ω

b(x)|u|^α|v|^βvΦ1− Z

Ω

c(x)|v|^β|u|^αuΦ2

and

(F,Φ) = ((f₁, f₂),(Φ₁,Φ₂)) = Z

Ω

f₁Φ₁+ Z

Ω

f₂Φ₂

We can write the operator A(u, v) as the sum of the two operators J2(v), J1(u), where

(J2(v),(Φ2)) = Z

Ω

|∇v|^q−2∇v∇Φ2 and (J1(u),(Φ1)) = Z

Ω

|∇u|^p−2∇u∇Φ1. OperatorsJ1andJ2are bounded, continuous, and strictly monotone; so their sum, the operatorA, will be the same. For the operatorB(u, v),

B(u, v) :W₀^1,p(Ω)×W₀^1,q(Ω)→L^p(Ω)×L^q(Ω)⊂W₀^−1,p0(Ω)×W₀^−1,q0(Ω), using Dominated convergence theorem and compact imbedding property [1] for the spaceW₀^1,p(Ω) inside the spaceL^p(Ω) and the spaceW₀^1,q(Ω) insideL^q(Ω), when Ω is a bounded domain ofR^N, we can prove that it is a strongly continuous operator.

To prove that let us assume thatv_n→^wv inW₀^1,q(Ω) andu_n→^w uinW₀^1,p(Ω). Then (un, vn)→(u, v) inL^p(Ω)×L^q(Ω). Also, (∇un,∇vn)→(∇u,∇v) inL^p(Ω)×L^q(Ω).

By the Dominated Convergence Theorem, we have:

a(x)|u_n|^p−2u_n →a(x)|u|^p−2u in L^p(Ω) d(x)|v_n|^q−2v_n→d(x)|v|^q−2v inL^q(Ω)

−b(x)|un|^α|vn|^βvn → −b(x)|u|^α|v|^βv in L^p(Ω)

−c(x)|v_n|^β|u_n|^αu_n→ −c(x)|v|^β|u|^αu in L^q(Ω).

Since

(B(un, vn)−B(u, v),(w1, w2))

= Z

Ω

a(x)(|u_n|^p−2u_n− |u|^p−2u)w₁+ Z

Ω

d(x)(|v_n|^q−2v_n− |v|^q−2v)w₂

− Z

b(x)(|u_n|^α|v_n|^βv_n− |u|^α|v|^βv)w₁− Z

Ω

c(x)(|v_n|^β|u_n|^αu_n− |v|^β|u|^αu)w₂, it follows that

kB(un, vn)−B(u, v)k

≤ ka(x)(|un|^p−2un− |u|^p−2u)kp+kd(x)(|vn|^q−2vn− |v|^q−2v)kq

+kb(x)(|un|^α|vn|^β+1− |u|^α|v|^β+1)kp+kc(x)(|un|^α+1|vn|^β− |u|^α+1|v|^β)kq →0. This proves that −B(u, v) is a strongly continuous operator. So A(u, v)−B(u, v) will be an operator satisfying the Mo-condition. Now, it remains to prove that A(u, v)−B(u, v) is a coercive operator:

|(A(u, v)−B(u, v),(u, v))|

= Z

Ω

|∇u|^p+ Z

|∇v|^q− Z

Ω

a(x)|u|^p− Z

d(x)|v|^q +

Z

Ω

b(x)|u|^α+1|v|^β+1+ Z

Ω

c(x)|u|^α+1|v|^β+1

≥ Z

Ω

|∇u|^p+ Z

Ω

|∇v|^q− 1 λ_p(a)

Z

Ω

|∇u|^p− 1 λ_q(d)

Z

Ω

|∇v|^q

= 1− 1

λp(a) Z

Ω

|∇u|^p+ 1− 1

λq(d) Z

Ω

|∇v|^q From (3.2), we deduce

(A(u, v)−B(u, v),(u, v))≥c(kuk^p_1,p+kvk^q_1,q) =c|(u, v)k_W^1,p

0 ×W₀^1,q

So that

hA(u, v)−B(u, v),(u, v)i → ∞ as k(u, v)k_W1,p

0 ×W₀^1,q → ∞.

This proves the coercive condition and so, the existence of a weak solution for

system (3.1).

4. Nonlinear systems defined on Rⁿ We consider the nonlinear system

−∆pu=a(x)|u|^p−2u−b(x)|u|^α|v|^βv+f,

−∆qv=−c(x)|u|^α|v|^βu+d(x)|v|^q−2v+g, lim

|x|→∞u= lim

|x|→∞v= 0 u, v >0

(4.1)

which is defined on R^N. We assume that 1≤ _N+1^2N < p, q < N and the coefficients a(x), b(x), c(x), d(x) are smooth positive functions such that

a(x), d(x)∈L^p/N(Rⁿ)∩L^∞(Rⁿ), α+ 1

p +β+ 1

q = 1, α+β+ 2< N, (4.2)

and

b(x)<(a(x))^α+1/p(d(x))^β+1/q

c(x)<(a(x))^α+1/p(d(x))^β+1/q (4.3) To prove our theorem, we need the following results which are studied in [12] and that we recall briefly: Let us introduce the Sobolev spaceD^1,p(R^N) defined as the completion ofC₀^∞(R^N) with respect to the norm

kukD^1,p=Z

R^N

|∇u|^p1/p .

It can be shown that

D^1,p(R^N) =

u∈L^{N−pN p} (R^N) :∇u∈(L^p(R^N))^N and that there existsk >0 such that for allu∈D^1,p(R^N),

kuk_LN p/(N−p) ≤Kkuk_D1,p(R^N). (4.4) Clearly, the spaceD^1,p(R^N) is a reflexive Banach space embedded continuously in the spaceL^{N p/(N−p)}(R^N).

Lemma 4.1. The eigenvalue problem

−∆pu=λa(x)|u|^p−2u inR^N

u(x)→0 as|x| → ∞ (4.5)

admits a positive principal eigenvalue Λ_a(p) which is associated with a positive eigenfunctionφ∈D^1,p(R^N); moreoverΛa(p)is characterized by

Λa(p) Z

R^N

a(x)|u|^p≤ Z

R^N

|∇u|^p, ∀u∈D^1,p(R^N) (4.6) Theorem 4.2. For (f, g)∈L^{N(p−1)+pN p} (R^N)×L^{N(q−1)+qN q} (R^N), there exists a weak solution(u, v)∈D^1,p(R^N)×D^1,q(R^N)for system (4.1)if the following conditions are satisfied:

Λ_p(a)>1, and Λ_q(d)>1. (4.7) Proof. By transforming the weak formulation for the system to the operator formu- lation, we will get the bounded operatorsA, B, F on the spaceD^1,p(R^N)×D^1,q(R^N) which take the same previous definitions in Theorem 3.1. To distinguish that: let us assume that (Φ₁,Φ₂) inD^1,p(R^N)×D^1,q(R^N), then applying H¨older inequality, we get

|(A(u, v),(Φ₁,Φ₂))|

≤ Z

R^N

|∇u|^p−1|∇Φ1|+ Z

R^N

|∇v|^q−1|∇Φ2|

≤Z

R^N

|∇u|^pp−1_p Z

R^N

|∇Φ1|^p1_p +Z

R^N

|∇v|^qq−1_q Z

R^N

|∇Φ2|^q1_q

=kuk^p−1_D1,pkΦ₁k_D1,p+kvk^q−1_D1,qkΦ₂k_D1,q

≤(kuk^p−1_D1,p+kvk^q−1_D1,q)(kΦ₁k_D1,p+kΦ₂k_D1,q)

= kuk^p−1_D1,p+kvk^q−1_D1,q

k(Φ₁,Φ₂)k_D1,p×D^1,q

For the operatorB(u, v), we have

|(B(u, v),(Φ1,Φ2))|

≤Z

(a(x))^Np_N^pZ

R^N

|u(x)|^{N−pN p (p−1)(N−p)}_{N p} Z

R^N

|Φ1|^{N−pN p N−p}_{N p} +Z

R^N

(d(x))^Nq_N^qZ

R^N

|v|^{N−qN q}

(q−1)(N−q)

N q Z

R^N

|Φ2|^{N−qN q N−q}_{N q} +Z

R^N

(b(x))^{α+β+2N α+β+2}_N Z

R^N

|u|^{N−pN p}

α(N−p) N p Z

R^N

|v|^{N−qN q}

(β+1)(N−q) N q

×Z

R^N

|Φ₁|^{N−pN p N−p}_{N p} +Z

R^N

(c(x))^{α+β+2N α+β+2}_N Z

R^N

|u|^{N−pN p (α+1)(N−p)}_{N p}

×Z

R^N

|v|^{N−qN q β(N−q)}_{N q} Z

R^N

|Φ₂|^{N−pN p N−q}_{N q}

≤k1kuk^p−1_D1,pkΦ1k_D1,p+k2kvk^q−1_D1,qkΦ2k_D1,q

+k₃kuk^α_D1,pkvk^β+1_D1,pkΦ₁k_D1,p+k₄kuk^α+1_D1,pkvk^β_D1,qkΦ₂k_D1,q

≤

k1kuk^p−1_D1,p+k2kvk^q−1_D1,q+k3kuk^α_D1,pkvk^β+1_D1,p+k4kuk^α+1_D1,pkvk^β_D1,q

× k(Φ1,Φ₂)kD^1,p×D^1,q,

this proves the boundedness of the operatorB(u, v). ForF, we have

|(F,Φ)|=|((f₁, f₂),(Φ₁,Φ₂))|

≤Z

R^N

(|f1|)^{N(p−1)+pN p N(p−1)+p}_{N p} Z

R^N

|Φ1|^{N−pN p N−p}_{N p}

+Z

R^N

(|f2|)^{N(q−1)+qN q N(q−1)+q}_{N q} Z

R^N

|Φ2|^{N−qN q N−q}_{N q}

≤

kf1k N p N(p−1)+p

+kf2k N q N(q−1)+q

k(Φ1,Φ₂)kD^1,p×D^1,q.

Now, the operatorA(u, v) =J1(u) +J2(v) is continuous and strictly monotone on D^1,p×D^1,q, since

(J1(u1)−J1(u2), u1−u2)≥(ku1k^p−1_D1,p− ku2k^p−1_D1,p)(ku1kD^1,p− ku2kD^1,q)>0, (J2(u1)−J2(u2), u1−u2)≥(ku1k^q−1_D1,q− ku2k^q−1_D1,q)(ku1k_D1,q− ku2k_D1,q)>0 For the operatorB(u, v), we can prove that it is a strongly continuous operator by using Dominated convergence theorem and continuous imbedding property for the spaceD^1,p(R^N)×D^1,q(R^N) intoL^{N−pN p} (R^N)×L^{N−qN q} (R^N): let us assume thatvn→^wv inD^1,q(R^N) andun→^wuinD^1,p(R^N). Then (un, vn)→(u, v) inL^p(R^N)×L^q(R^N) and (∇un,∇vn) → (∇u,∇v) in L^p(R^N)×L^q(R^N). Now, the sequence (un) is bounded in D^1,p(R^N), then it is containing a subsequence again denoted by (un) converges strongly touinL^{N−pN p} (Br₀) for any bounded ballBr₀ ={x∈R^N :kxk ≤ r₀}. Similarly (v_n) converges strongly tovinL^{N−qN q} (B_r0). Sinceu_n, u∈L^{N−pN p} (B_r0)

andv_n, v∈L^{N−qN q} (B_r0). Then using the dominated convergence theorem, we have ka(x)(|un|^p−2un− |u|^p−2u)k N p

N(p−1)+p →0, (4.8)

kd(x)(|vn|^q−2v_n− |v|^q−2v)k ^{N q}

N(q−1)+q →0, (4.9)

kb(x)(|u_n|^α−1|v_n|^β+1u_n− |u|^α−1|v|^β+1u)k N p

N(p−1)+p →0, (4.10)

kc(x)(|un|^α+1|vn|^β−1un− |u|^α+1|v|^β−1u)k N q

N(q−1)+q →0. (4.11) Then

kB(un, v_n)−B(u, v)kD^1,p(B_r0)×D^1,q(B_r0)

≤ ka(x)(|un|^p−2un− |u|^p−2u)k N p

N(p−1+p) +kd(x)(|vn|^q−2vn− |v|^q−2v)k N q N(q−1+q)

+kb(x)(|un|^α|vn|^β+1un− |u|^α|v|^β+1v)k N p N(p−1)+p

+kc(x)(|un|^α+1|vn|^β−1u_n− |u|^α+1|v|^β−1v)k N q

N(q−1)+q →0. It remains to study the norm

kB(un, vn)−B(u, v)k_D1,p(R^N−Br0)×D^1,q(R^N−Br0)

It is sufficient to study the norms in the inequalities (4.8)–(4.11) and try to make it as small as possible. We will study the norm in (4.8) only because the others will be the same.

Since, (un) converges weakly in the space D^1,p(R^N), using Sobelev inequal- ity, (u_n) will be bounded in the space L^{N−pN p} (R^N), so |un|^p−1 will be bounded in L^{N(p−1)+pN p} (R^N−Br₀) and (|un|^p−2un−|u|^p−2u) is bounded inL^{N(p−1)+pN p} (R^N−Br₀).

Since,a(x)∈L^Np(R^N), we can make the integralR

(R^N−Br0)|a(x)|^Np as small as possible by choosing r0 big as possible, this means that there exists r0 > 0 such that

ka(x)(|un|^p−2un− |u|^p−2u)k

L

N p

N(p−1)+p(R^N−Br0)

<

4.M = 4 for alln≥N₀,r≥r₀. Since

ka(x)(|un|^p−2un− |u|^p−2u)k

L

N p N(p−1)+p(R^N)

=ka(x)(|u_n|^p−2u_n− |u|^p−2u)k

L

N p

N(p−1)+p(Br0)

+ka(x)(|un|^p−2un− |u|^p−2u)k

L

N p

N(p−1)+p(R^N−Br0)

,

it follows that

ka(x)(|un|^p−2un− |u|^p−2u)k

L

N p N(p−1)+p(R^N)

→0. By repeating the previous steps on the remaining terms in

kB(un, vn)−B(u, v)k_D1,p(R^N)×D^1,q(R^N),

we can prove that this norm tending strongly to zero and then the operatorB(u, v) is strongly continuous. It remains to justify that the operator A(u, v)−B(u, v) is

a coercive operator. From (4.3), (4.6) and (4.7), we obtain (A(u, v)−B(u, v),(u, v))

= Z

R^N

|∇u|^p+ Z

R^N

|∇v|^q− Z

R^N

a(x)|u|^p− Z

R^N

d(x)|v|^q +

Z

b(x)|u|^α+1|v|^β+1+ Z

b(x)|u|^α+1|v|^β+1

≥ Z

R^N

|∇u|^p+ Z

R^N

|∇v|^q− 1 Λa(p)

Z

R^N

|∇u|^p− 1 Λd(q)

Z

R^N

|∇v|^q

= 1− 1

Λ_a(p) Z

R^N

|∇u|^p+ 1− 1

Λ_d(q) Z

R^N

|∇v|^q

> c

kuk^p_D1,p+kvk^q_D1,q .

So that

(A(u, v)−B(u, v),(u, v))→ ∞ as k(u, v)k_D1.p×D^1,q → ∞

The coercive condition for the operator completes the proof of the existence of a

weak solution for system (4.1).

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Eada A. El-Zahrani

Mathematics Department, Faculty of Science for Girls,, Dammam, P. O. Box 838, Pin- code 31113, Saudi Arabia

Hassan M. Serag

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt

E-mail address:[email protected]