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| ∫