Existence of Solutions of a Nonlocal Elliptic System via Galerkin Method

and Applied Analysis 3 x 1. That is, sin πx > 0 in 0, 1 but, for η large enough, the corresponding solution is not positive. This contrasts with the local equation −u′′ x ηu x f x , 0 < x < 1, u 0 u 1 0, η > 0, 1.8 for which it is very well known, see 14 , that, for all η > 0, function u > 0 in 0, 1 whenever f > 0 in 0, 1 . Remark 1.1. We have to point out that the lack of a general maximum principle seems to be characteristic of integrodifferential operator. Indeed, in 15 , the authors consider a noncooperative system, arisen in the classical FitzHugh-Nagumo systems, which serves as a model for nerve conduction. More precisely, it is studied the system −Δu x f x, u − v, x ∈ Ω, −Δv x δu − γv, x ∈ Ω, u x , v x > 0, x ∈ Ω, u x v x 0, x ∈ ∂Ω, 1.9 where δ, γ > 0 are constants and f x, u is a given function. Taking B ≡ δ −Δ γ −1, under Dirichlet boundary condition, problem 1.9 is equivalent to the integrodifferential problem −Δu x Bu f x, u , x ∈ Ω, u x 0, x ∈ ∂Ω. 1.10 Consider now the problem −Δu x Bu − λu f x , x ∈ Ω, u x 0, x ∈ ∂Ω, 1.11 with f ∈ L2 Ω and f ≥ 0 in Ω. Let λ1 be the first eigenvalue of operator −Δ in the space H1 0 Ω , and assume that λ1 > √ δ − γ . Then, for all λ ∈ 2 √ δ − γ, λ1 δ/ γ λ1 , problem 1.11 satisfies a maximum principle, that is, under the above assumptions, the solution u of 1.11 satisfies u ≥ 0 a.e in Ω. See 15 for the proof of this result. After that, it is proved in 16 , by using semigroup theory, that this maximum principle does not hold for all λ < 2 √ δ−γ . Indeed, the approach used in 16 may be used to prove that a general maximum principle for the problem 1.6 is not valid. In view of this, the method of suband supersolution should be used carefully by considering a relation between the growth of the nonlinearity and the parameters of the problem. 4 Abstract and Applied Analysis Remark 1.2. It is worthy to remark that problem 1.1 has no variational structure even in the scalar case. So the most usual variational techniques cannot be used to study it. To attack problem 1.1 , we will use the Galerkin method through the following version of the Brouwer fixed-Point Theorem whose proof may be found in Lions 17, Lemma 4.3 . Proposition 1.3. Let F : R → R be a continuous function such that 〈F ξ , ξ〉 > 0 if |ξ| r, 1.12 for some r > 0, where 〈·, ·〉 is the Euclidian Scalar product and | · | 〈·, ·〉 is the corresponding Euclidian norm in R. Then, there exists ξ0 ∈ R, |ξ0| ≤ r such that F ξ0 0. 2. A Sublinear Problem In this section, we consider the problem −Δu x λ ∫ Ω v ( y ) dy u x v x , x ∈ Ω, −Δv x λ ∫ Ω u ( y ) dy u x v x , x ∈ Ω, u x , v x > 0, x ∈ Ω, u x v x 0, x ∈ ∂Ω. 2.1 Here, λ is a real parameter and α, β, γ, δ are positive constants whose properties will be precised later. In order to use Proposition 1.3, we have to introduce a suitable setup. First of all, we consider an orthonormal Hilbertian basis B {φ1, φ2, . . .} in H1 0 Ω whose norm is the usual one ‖u‖ ∫ Ω ∣∣∇u(y)∣∣2 dy, ∀u ∈ H1 0 Ω . 2.2 Next, let Vm be the finite dimensional vector space Vm [ φ1, . . . , φm ] ⊂ B, 2.3 equipped with the norm induced by the one inH1 0 Ω . Thus, if u ∈ Vm, there is a unique ξ ξ1, . . . , ξm ∈ R such that u m ∑ j 1 ξjφj , 2.4 Abstract and Applied Analysis 5 and, as a consequence, ‖u‖ |ξ|. 2.5 So, the spaces Vm and R are isomorphic and isometric by Vm ←→ R, u m ∑ j 1 ξjφj ←→ ξ ξ1, . . . , ξm . 2.6and Applied Analysis 5 and, as a consequence, ‖u‖ |ξ|. 2.5 So, the spaces Vm and R are isomorphic and isometric by Vm ←→ R, u m ∑ j 1 ξjφj ←→ ξ ξ1, . . . , ξm . 2.6 From now on, we identify, with no additional comments, u↔ ξ via this isometry. In order to obtain a nontrivial solution of problem 2.1 , let > 0 be a constant and consider the auxiliary problem −Δu x λ ∫ Ω v p ( y ) dy u α x v β x , x ∈ Ω, −Δv x λ ∫ Ω u q ( y ) dy u γ x v δ x , x ∈ Ω, u x , v x > 0, x ∈ Ω, u x v x 0, x ∈ ∂Ω. 2.7 Theorem 2.1. Assume thatΩ is a boundedC2 domain ofR and the constants p, q, α, β, γ, δ ∈ 0, 1 . Then, for all λ < 0, problem 2.1 has at least one solution in C2 Ω ∩ C Ω × C2 Ω ∩ C Ω . Proof. First of all, we consider a map F,G : R × R → R × R, F,G F1, . . . , Fm,G1, . . . , Gm , defined, for all i 1, . . . , m, as Fi ( ξ, η ) ∫ Ω ∇u∇φi λ ∫ Ω v p ∫ Ω φi − ∫ Ω u φi − ∫ Ω v φi − ∫ Ω φi, Gi ( ξ, η ) ∫ Ω ∇v∇φi λ ∫ Ω u q ∫ Ω φi − ∫ Ω u φi − ∫ Ω v φi, 2.8 where we are identifying u, v ∈ Vm × Vm, u ∑m j 1 ξjφj , v ∑m j 1 ηjφj , with ξ, η ∈ R × R , ξ ξ1, . . . , ξm , η η1, . . . , ηm . Now, we have that, for all i 1, . . . , m, the following equations hold: Fi ( ξ, η ) · ξi ∫ Ω ∇u · ∇(ξiφi) λ ∫ Ω v p ∫ Ω ξiφi − ∫ Ω u α ( ξiφi ) − ∫ Ω v β ( ξiφi ) − ∫ Ω ξiφi, Gi ( ξ, η ) · ηi ∫ Ω ∇v · ∇(ηiφi) λ ∫ Ω u q ∫ Ω ηiφi − ∫ Ω u γ ( ηiφi ) − ∫ Ω v δ ( ηiφi ) . 2.9 6 Abstract and Applied Analysis Therefore, 〈 F,G ( ξ, η ) , ( ξ, η )〉 ∫ Ω |∇u| ∫


Introduction
This paper is devoted to the study of the nonlocal elliptic system −Δu x a x Ω b y v p y dy g u x , v x , x ∈ Ω,

1.1
Here, Ω ⊂ R N , N ≥ 2, is a bounded smooth domain, p, q are positive numbers, a, b, c, d ∈ C Ω , and the nonlinearities g and h will be defined later.
The one-dimensional counterpart of this problem has been considered by Cabada et al. in 1 . There the authors, by using dual variational methods and Leray-Schauder degree, with p q 1, b ≡ d ≡ 1, and a c a real parameter, and under suitable assumptions on the nonlinear functions g u, v ≡ g v and h u, v ≡ h u , show the existence of solution not necessarily positive depending upon the parameter a.
In this case, we present a different point of view from the one used in 1 . We note that, among other things, we assume N ≥ 2.
Motivated by its many applications and the richness of the methods employed to solve it, this kind of problems has been studied by different authors when only one equation is considered, see, among others, 2-9 . Indeed, there is a lot of phenomena that may be modeled by equations of the form where B is a nonlocal operator which, in some applications, is written in the form See 10-13 for some surveys on these equations. In particular, steady-state solutions deliver us to elliptic equations such as which, in several cases, have a behaviour quite different from the local one One of the most significant differences between these two types of problems is the nonexistence, in some particular cases, of maximum principles. For instance, Allegretto and Barabanova 4 consider the one-dimensional problem It is not difficult to verify that the explicit solution of this problem is given by the expression So, when the values of the positive parameter η are small, the solution is positive. However, if η is large enough, function u becomes negative near the end points x 0 and x 1. That is, sin πx > 0 in 0, 1 but, for η large enough, the corresponding solution is not positive.
This contrasts with the local equation for which it is very well known, see 14 , that, for all η > 0, function u > 0 in 0, 1 whenever f > 0 in 0, 1 .
We have to point out that the lack of a general maximum principle seems to be characteristic of integrodifferential operator. Indeed, in 15 , the authors consider a noncooperative system, arisen in the classical FitzHugh-Nagumo systems, which serves as a model for nerve conduction. More precisely, it is studied the system where δ, γ > 0 are constants and f x, u is a given function. Taking B ≡ δ −Δ γ −1 , under Dirichlet boundary condition, problem 1.9 is equivalent to the integrodifferential problem −Δu x Bu f x, u , x ∈ Ω, u x 0, x ∈ ∂Ω.

1.10
Consider now the problem Let λ 1 be the first eigenvalue of operator −Δ in the space H 1 0 Ω , and assume that Then, for all λ ∈ 2 √ δ − γ, λ 1 δ/ γ λ 1 , problem 1.11 satisfies a maximum principle, that is, under the above assumptions, the solution u of 1.11 satisfies u ≥ 0 a.e in Ω. See 15 for the proof of this result.
After that, it is proved in 16 , by using semigroup theory, that this maximum principle does not hold for all λ < 2 √ δ − γ. Indeed, the approach used in 16 may be used to prove that a general maximum principle for the problem 1.6 is not valid. In view of this, the method of sub-and supersolution should be used carefully by considering a relation between the growth of the nonlinearity and the parameters of the problem.

Remark 1.2.
It is worthy to remark that problem 1.1 has no variational structure even in the scalar case. So the most usual variational techniques cannot be used to study it.
To attack problem 1.1 , we will use the Galerkin method through the following version of the Brouwer fixed-Point Theorem whose proof may be found in Lions 17, Lemma 4.3 .

A Sublinear Problem
In this section, we consider the problem

2.1
Here, λ is a real parameter and α, β, γ, δ are positive constants whose properties will be precised later.
In order to use Proposition 1.3, we have to introduce a suitable setup. First of all, we consider an orthonormal Hilbertian basis B {ϕ 1 , ϕ 2 , . . .} in H 1 0 Ω whose norm is the usual one Next, let V m be the finite dimensional vector space Abstract and Applied Analysis 5 and, as a consequence, So, the spaces V m and R m are isomorphic and isometric by From now on, we identify, with no additional comments, u ↔ ξ via this isometry.
In order to obtain a nontrivial solution of problem 2.1 , let > 0 be a constant and consider the auxiliary problem

2.7
Theorem 2.1. Assume that Ω is a bounded C 2 domain of R N and the constants p, q, α, β, γ, δ ∈ 0, 1 . Then, for all λ < 0, problem 2.1 has at least one solution in Proof. First of all, we consider a map F, G : . . , η m . Now, we have that, for all i 1, . . . , m, the following equations hold: Abstract and Applied Analysis Therefore,

2.10
Denoting as x, y 2 |x| 2 |y| 2 for all x, y ∈ R m , using the isometry between V m and R m , and the inequalities of Hölder, Poincaré, and Sobolev, we arrive at the following estimations: and, in a similar way, We note that the positive constant C depends on Ω, but it does not depend on the rest of the parameters involved in problem 2.7 .
Abstract and Applied Analysis 7 Using the previous estimations I 1 -I 6 together with the fact that λ < 0, we deduce that

2.11
Now, since 0 < p, q, α, β, γ, δ < 1, there is r > 0 such that In view of Proposition 1.3, there exists r > 0 that does not depend on m, and a pair u m , v m ∈ V m × V m , such that u m , v m ≤ r, and satisfies the following equalities for all i 1, . . . , m :

2.13
So, for all ϕ ∈ V m , it is satisfied that

2.14
In view of the boundedness of the approximate solutions u m , v m in H 1 0 Ω , we have, perhaps for subsequences, that u m u and v m v in H 1 0 Ω . Fixing k < m and making m → ∞ in the last two equalities, we obtain, after using Sobolev immersions, that the following equalities hold for all ϕ ∈ V k :

2.15
Since k is arbitrary, the last two identities are valid for all ϕ ∈ H 1 0 Ω . Consequently, u , v ∈ H 1 0 Ω × H 1 0 Ω is a weak solution of the auxiliary problem 2.7 .

Abstract and Applied Analysis
Now, since λ < 0 and > 0, we have, from 2.7 , that −Δu ≥ and −Δv ≥ 0 on Ω. This fact, together with the Dirichlet boundary conditions, says that u , v > 0 on Ω. In consequence, the following equalities hold for all ϕ ∈ H 1 0 Ω :

2.16
That is to say, u , v is a weak solution of problem

2.17
In particular, it satisfies the following inequalities:

2.18
Let u α , v δ > 0 be the unique solution of the problem

2.19
We point out that the existence and uniqueness of the solutions of each of the above equations follow from 18, 19 because 0 < α, δ < 1.
In the sequel, we use the following comparison result, due to Ambrosetti, Brezis, and Cerami.

2.20
Then, w x ≥ v x for all x ∈ Ω.
So we conclude that u ≥ u α > 0 and v ≥ v δ > 0 in Ω.
Taking limits on both members of 2.16 and 2.6 as → 0 , we deduce that u u and v v, for some u, v ∈ H 1 0 Ω such that u, v > 0 in Ω. Proceeding as before, by using Sobolev embeddings and elliptic regularity, we conclude that u, v is a classical solution of the system 2.1 .

Remark 2.3.
We note that, from the fact that problem 1.6 is a one-dimensional particular case of problem 2.1 , when λ > 0, we cannot ensure that, in general, the problem 2.1 has a positive solution in Ω.

Remark 2.4.
We should point out that we may consider a more general system than 2.1 . To be more precise, we may consider a system like

2.21
where −Δ m u − div |∇u| m−2 ∇u and −Δ n u − div |∇u| n−2 ∇u are, respectively, the m-Laplacian and n-Laplacian. Although the proof of the existence of solution follows similar ideas as those used in Theorem 2.1, the calculations are more complicated because we have to work with Schauder's basis in W 1,m 0 Ω and W 1,n 0 Ω and these spaces do not enjoy a Hilbert space structure.
If we would like to dare a little more, we may consider a system like

A Singular Problem
The Galerkin method may also be used to attack a singular version of the problem 2.1 . More precisely, let us consider a simple version of a singular problem as with α, δ > 0.
We should point out that other combinations of u and v may be considered, including convection terms like |∇u| γ , γ > 0. More precisely, we may consider problems like

3.3
However, for the sake of simplicity and to illustrate the method, we restrict our discussion to the problem 3.1 .
To approach problem 3.1 , we consider a nonsingular perturbation as with > 0, to obtain approximate solutions u , v . In this way, we avoid the singular term.
We have the following result.
Proof. Reasoning as in the proof of Theorem 2.1 we obtain, for each > 0, a solution u , v of problem 3.4 . So, since λ < 0, we obtain

3.5
In view of the maximum principle, we obtain that u > 0 in Ω and v ≥ 0 in Ω. Consequently,

3.6
If v ≡ 0, we obtain, in view of 3.6 , that Ω u q y dy 0, which contradicts the fact that u > 0 in Ω. Thus, v / ≡ 0 and, because v ≥ 0, the maximum principle gives v > 0 in Ω. The solution of problem 3.1 will be obtained by studying the limit when → 0. Thus, we may suppose that 0 < < 1. In view of this, we obtain from 3.6 that

3.7
Let ω be the unique positive solution of
Using the Galerkin method and reasoning as in the proof of Theorem 2.1, we deduce that the approximate solutions u , v are uniformly bounded in H 1 0 Ω with respect to 0 < < 1. In consequence, we conclude that u u and v v in H 1 0 Ω in the weak sense. Hence, in view of 3.8 , we obtain 3.9 Invoking again the maximum principle, we get ω > 0 in Ω. Since u > ω in Ω, we conclude that u ≥ ω > 0 in Ω. Reasoning as before, v > 0 in Ω, and u, v is a classical solution of problem 3.1 , which finishes the proof of this result.

On a Superlinear Problem
At last we will make some remarks on a superlinear problem. In order to simplify the exposition, let us consider the one equation case

4.3
As before, let us consider F : R m → R m , F F 1 , . . . , F m , defined, for all i 1, . . . , m, as where we are using the previous identifications. For all i 1, . . . , m, the following equations hold