Existence of Solutions of a Nonlocal Elliptic System via Galerkin Method

Abstract

By means of the Galerkin method and by using a suitable version of the Brouwer fixed-point theorem, we establish the existence of at least one positive solution of a nonlocal elliptic N-dimensional system coupled with Dirichlet boundary conditions.

1. Introduction

This paper is devoted to the study of the nonlocal elliptic system−Δ𝑢(𝑥)+𝑎(𝑥)_Ω𝑏𝑦𝑣^𝑝𝑦(𝑢(𝑥)(𝑥(𝑥)(𝑥)𝑑𝑦=𝑔,𝑣)),𝑥∈Ω,−Δ𝑣+𝑐_Ω𝑑𝑦𝑢^𝑞𝑦(𝑢(𝑥)(𝑥𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥)𝑑y=ℎ,𝑣)),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,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 form𝑢_𝑡−Δ𝑢=𝑓(𝑥,𝑢,𝐵(𝑢)), (1.2) where 𝐵 is a nonlocal operator which, in some applications, is written in the form𝐵(𝑢)=_Ω𝑏(𝑥)[𝑢(𝑥)]^𝛽𝑑𝑥. (1.3)

See [10–13] for some surveys on these equations.

In particular, steady-state solutions deliver us to elliptic equations such as(𝑥)((𝑥)(𝑢𝑢(𝑥)−Δ𝑢=𝑓𝑥,𝑢,𝐵)),𝑥∈Ω,=0,𝑥∈𝜕Ω, (1.4) which, in several cases, have a behaviour quite different from the local one(𝑥)((𝑥𝑢(𝑥)−Δ𝑢=𝑓𝑥,𝑢)),𝑥∈Ω,=0,𝑥∈𝜕Ω. (1.5)

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−𝑢^′′(𝑥)+𝜂¹₀𝑢𝑦()𝑢(0)(1)𝑑𝑦=sin𝜋𝑥,0<𝑥<1,=𝑢=0,𝜂>0. (1.6)

It is not difficult to verify that the explicit solution of this problem is given by the expression𝑢(𝑥)=12𝜂𝑥²−𝑥𝜋³+()12+𝜂sin𝜋𝑥𝜋². (1.7)

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−𝑢^′′(𝑥)(𝑥)(𝑥)𝑢(0)(1)+𝜂𝑢=𝑓,0<𝑥<1,=𝑢=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 (𝑥)()(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥)−Δ𝑢=𝑓𝑥,𝑢−𝑣,𝑥∈Ω,−Δ𝑣=𝛿𝑢−𝛾𝑣,𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=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 (𝑥)()𝑢(𝑥)−Δ𝑢+𝐵𝑢=𝑓𝑥,𝑢,𝑥∈Ω,=0,𝑥∈𝜕Ω. (1.10)
Consider now the problem (𝑥)(𝑥)𝑢(𝑥)−Δ𝑢+𝐵𝑢−𝜆𝑢=𝑓,𝑥∈Ω,=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 𝐻¹₀(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 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].

Proposition 1.3. Let 𝐹∶ℝ^𝑚→ℝ^𝑚 be a continuous function such that 𝐹(𝜉),𝜉>0if𝜉=𝑟, (1.12) for some 𝑟>0, where ⟨⋅,⋅⟩ is the Euclidian Scalar product and |⋅|=⟨⋅,⋅⟩^1/2 is the corresponding Euclidian norm in ℝ^𝑚. Then, there exists 𝜉₀∈ℝ^𝑚,|𝜉₀|≤𝑟 such that 𝐹(𝜉₀)=0.

2. A Sublinear Problem

In this section, we consider the problem−Δ𝑢(𝑥)+𝜆_Ω𝑣^𝑝𝑦𝑑𝑦=𝑢^𝛼(𝑥)+𝑣^𝛽(𝑥)(𝑥),𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^𝑞𝑦𝑑𝑦=𝑢^𝛾(𝑥)+𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=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 𝐻¹₀(Ω) whose norm is the usual one‖𝑢‖²=_Ω∇𝑢𝑦²𝑑𝑦,∀𝑢∈𝐻¹₀(Ω). (2.2)

Next, let 𝕍_𝑚 be the finite dimensional vector space𝕍_𝑚=𝜑₁,…,𝜑_𝑚⊂𝔹, (2.3)

equipped with the norm induced by the one in 𝐻¹₀(Ω).

Thus, if 𝑢∈𝕍_𝑚, there is a unique 𝜉=(𝜉₁,…,𝜉_𝑚)∈ℝ^𝑚 such that𝑢=^𝑚_𝑗=1𝜉_𝑗𝜑_𝑗, (2.4)

and, as a consequence,‖𝑢‖²=𝜉². (2.5)

So, the spaces 𝕍_𝑚 and ℝ^𝑚 are isomorphic and isometric by𝕍_𝑚⟷ℝ^𝑚,𝑢=^𝑚_𝑗=1𝜉_𝑗𝜑_𝑗⟷𝜉=𝜉₁,…,𝜉_𝑚. (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 𝜖>0 be a constant and consider the auxiliary problem−Δ𝑢(𝑥)+𝜆_Ω𝑣^+𝑝𝑦𝑑𝑦=𝑢^+𝛼(𝑥)+𝑣^+𝛽(𝑥)(𝑥)+𝜖,𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^+𝑞𝑦𝑑𝑦=𝑢^+𝛾(𝑥)+𝑣^+𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω. (2.7)

Theorem 2.1. Assume that Ω is a bounded 𝐶² 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 (𝐶²(Ω)∩𝐶(Ω))×(𝐶²(Ω)∩𝐶(Ω)).

Proof. First of all, we consider a map (𝐹,𝐺)∶ℝ^𝑚×ℝ^𝑚→ℝ^𝑚×ℝ^𝑚,(𝐹,𝐺)=(𝐹₁,…,𝐹_𝑚,𝐺₁,…,𝐺_𝑚), defined, for all 𝑖=1,…,𝑚, as 𝐹_𝑖𝜉,𝜂=_Ω∇𝑢∇𝜑_𝑖+𝜆_Ω𝑣^+𝑝_Ω𝜑_𝑖−_Ω𝑢^+𝛼𝜑_𝑖−_Ω𝑣^+𝛽𝜑_𝑖−𝜖_Ω𝜑_𝑖,𝐺_𝑖𝜉,𝜂=_Ω∇𝑣∇𝜑_𝑖+𝜆_Ω𝑢^+𝑞_Ω𝜑_𝑖−_Ω𝑢^+𝛾𝜑_𝑖−_Ω𝑣^+𝛿𝜑_𝑖, (2.8)where we are identifying (𝑢,𝑣)∈𝕍_𝑚×𝕍_𝑚,𝑢=∑^𝑚_𝑗=1𝜉_𝑗𝜑_𝑗,𝑣=∑^𝑚_𝑗=1𝜂_𝑗𝜑_𝑗, with (𝜉,𝜂)∈ℝ^𝑚×ℝ^𝑚,  𝜉=(𝜉₁,…,𝜉_𝑚),  𝜂=(𝜂₁,…,𝜂_𝑚).
Now, we have that, for all 𝑖=1,…,𝑚, the following equations hold: 𝐹_𝑖𝜉,𝜂⋅𝜉_𝑖=_Ω∇𝑢⋅∇𝜉_𝑖𝜑_𝑖+𝜆_Ω𝑣^+𝑝_Ω𝜉_𝑖𝜑_𝑖−_Ω𝑢^+𝛼𝜉_𝑖𝜑_𝑖−_Ω𝑣^+𝛽𝜉_𝑖𝜑_𝑖−𝜖_Ω𝜉_𝑖𝜑_𝑖,𝐺_𝑖𝜉,𝜂⋅𝜂_𝑖=_Ω∇𝑣⋅∇𝜂_𝑖𝜑_𝑖+𝜆_Ω𝑢^+𝑞_Ω𝜂_𝑖𝜑_𝑖−_Ω𝑢^+𝛾𝜂_𝑖𝜑_𝑖−_Ω𝑣^+𝛿𝜂_𝑖𝜑_𝑖. (2.9)
Therefore, (𝐹,𝐺)𝜉,𝜂,𝜉,𝜂=_Ω|∇𝑢|²+_Ω|∇𝑣|²+𝜆_Ω𝑣^+𝑝_Ω𝑢−_Ω𝑢^+𝛼𝑢−_Ω𝑣^+𝛽𝑢−𝜖_Ω𝑢+𝜆_Ω𝑢^+𝑞_Ω𝑣−_Ω𝑢^+𝛾𝑣−_Ω𝑣^+𝛿𝑣. (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: _Ω𝑣^+𝑝_Ω𝑢≤𝐶_Ω(|𝑣|^𝑝)^2/𝑝𝑝/2⋅‖𝑢‖≤𝐶‖(𝑢,𝑣)‖^𝑝+1, (𝐼₁)_Ω𝑣^+𝛽𝑢≤_Ω|𝑣|^2𝛽1/2|𝑢|₂≤𝐶‖(𝑢,𝑣)‖^𝛽+1, (𝐼₂) and, in a similar way, _Ω𝑢^+𝛼𝑢≤𝐶‖(𝑢,𝑣)‖^𝛼+1, (𝐼₃)_Ω𝑢^+𝑞_Ω𝑣≤𝐶‖(𝑢,𝑣)‖^𝑞+1, (𝐼₄)_Ω𝑢^+𝛾𝑣≤‖(𝑢,𝑣)‖^𝛾+1, (𝐼₅)_Ω𝑣^+𝛿𝑣≤𝐶‖(𝑢,𝑣)‖^𝛿+1. (𝐼₆)
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).
Using the previous estimations (𝐼₁)–(𝐼₆) together with the fact that 𝜆<0, we deduce that (𝐹,𝐺)𝜉,𝜂,𝜉,𝜂≥‖(𝑢,𝑣)‖²+𝜆𝐶‖(𝑢,𝑣)‖^𝑝+1−𝐶‖(𝑢,𝑣)‖^𝛽+1−𝜖𝐶‖(𝑢,𝑣)‖+𝜆𝐶‖(𝑢,𝑣)‖^𝑞+1−𝐶‖(𝑢,𝑣)‖^𝛾+1−𝐶‖(𝑢,𝑣)‖^𝛿+1−𝐶‖(𝑢,𝑣)‖^𝛾+1−𝐶‖(𝑢,𝑣)‖^𝛼+1. (2.11)
Now, since 0<𝑝,𝑞,𝛼,𝛽,𝛾,𝛿<1, there is 𝑟>0 such that (𝐹,𝐺)𝜉,𝜂,𝜉,𝜂>0if𝜉,𝜂>𝑟. (2.12)
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∶_Ω∇𝑢_𝑚∇𝜑_𝑖+𝜆_Ω𝑣⁺_𝑚^𝑝_Ω𝜑_𝑖−_Ω𝑢⁺_𝑚^𝛼𝜑_𝑖−_Ω𝑣⁺_𝑚^𝛼𝜑_𝑖−𝜖_Ω𝜑_𝑖=0,_Ω∇𝑣_𝑚∇𝜑_𝑖+𝜆_Ω𝑢⁺_𝑚^𝑞_Ω𝜑_𝑖−_Ω𝑢⁺_𝑚^𝛾𝜑_𝑖−_Ω𝑣⁺_𝑚^𝛿𝜑_𝑖=0. (2.13)
So, for all 𝜑∈𝕍_𝑚, it is satisfied that _Ω∇𝑢_𝑚∇𝜑+𝜆_Ω𝑣⁺_𝑚^𝑝_Ω𝜑−_Ω𝑢⁺_𝑚^𝛼𝜑−_Ω𝑣⁺_𝑚^𝛼𝜑−𝜖_Ω𝜑=0,_Ω∇𝑣_𝑚∇𝜑+𝜆_Ω𝑢⁺_𝑚^𝑞_Ω𝜑−_Ω𝑢⁺_𝑚^𝛾𝜑−_Ω𝑣⁺_𝑚^𝛿𝜑=0. (2.14)
In view of the boundedness of the approximate solutions 𝑢_𝑚,𝑣_𝑚 in 𝐻¹₀(Ω), we have, perhaps for subsequences, that 𝑢_𝑚⇀𝑢_𝜖 and 𝑣_𝑚⇀𝑣_𝜖 in 𝐻¹₀(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: _Ω∇𝑢_𝜖∇𝜑+𝜆_Ω𝑣⁺_𝜖^𝑝_Ω𝜑−_Ω𝑢⁺_𝜖^𝛼𝜑−_Ω𝑣⁺_𝜖^𝛼𝜑−𝜖_Ω𝜑=0,_Ω∇𝑣_𝜖∇𝜑+𝜆_Ω𝑢⁺_𝜖^𝑞_Ω𝜑−_Ω𝑢⁺_𝜖^𝛾𝜑−_Ω𝑣⁺_𝜖^𝛿𝜑=0. (2.15)
Since 𝑘 is arbitrary, the last two identities are valid for all 𝜑∈𝐻¹₀(Ω). Consequently, (𝑢_𝜖,𝑣_𝜖)∈𝐻¹₀(Ω)×𝐻¹₀(Ω) is a weak solution of the auxiliary problem (2.7).
Now, since 𝜆<0 and 𝜖>0, we have, from (2.7), that −Δ𝑢_𝜖≥𝜖 and −Δ𝑣_𝜖≥0 on Ω. This fact, together with the Dirichlet boundary conditions, says that 𝑢_𝜖,𝑣_𝜖>0 on Ω. In consequence, the following equalities hold for all 𝜑∈𝐻¹₀(Ω): _Ω∇𝑢_𝜖∇𝜑+𝜆_Ω𝑣^𝑝_𝜖Ω𝜑−_Ω𝑢^𝛼𝜑−_Ω𝑣^𝛼𝜑−𝜖_Ω𝜑=0,_Ω∇𝑣_𝜖∇𝜑+𝜆_Ω𝑢^𝑞_𝜖Ω𝜑−_Ω𝑢^𝛾_𝜖𝜑−_Ω𝑣^𝛿_𝜖𝜑=0. (2.16)
That is to say, (𝑢_𝜖,𝑣_𝜖) is a weak solution of problem −Δ𝑢(𝑥)+𝜆_Ω𝑣^𝑝𝑦𝑑𝑦=𝑢^𝛼(𝑥)+𝑣^𝛽(𝑥)(𝑥)+𝜖,𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^𝑞𝑦𝑑𝑦=𝑢^𝛾(𝑥)+𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω. (2.17)
In particular, it satisfies the following inequalities: −Δ𝑢_𝜖(𝑥)≥𝑢^𝛼_𝜖(𝑥),𝑥∈Ω,−Δ𝑣_𝜖(𝑥)≥𝑣^𝛿_𝜖(𝑥)𝑢,𝑥∈Ω,_𝜖(𝑥),𝑣_𝜖(𝑥)𝑢>0,𝑥∈Ω,_𝜖(𝑥)=𝑣_𝜖(𝑥)=0,𝑥∈𝜕Ω. (2.18)
Let 𝑢_𝛼,𝑣_𝛿>0 be the unique solution of the problem −Δ𝑢(𝑥)=𝑢^𝛼(𝑥)(𝑥),𝑥∈Ω,−Δ𝑣=𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω. (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.
Lemma 2.2. [20, Lemma 3.3]. Assume that 𝑓(𝑡) is a continuous function such that 𝑡⁻¹𝑓(𝑡) is decreasing for 𝑡>0. Let 𝑣 and 𝑤 satisfy (𝑥)(𝑣(𝑥𝑣(𝑥)𝑣(𝑥)(𝑥)(𝑤(𝑥𝑤(𝑥)𝑤(𝑥)−Δ𝑣≤𝑓)),𝑥∈Ω,>0,𝑥∈Ω,=0,𝑥∈𝜕Ω,−Δ𝑤≥𝑓)),𝑥∈Ω,>0,𝑥∈Ω,=0,𝑥∈𝜕Ω. (2.20) Then, 𝑤(𝑥)≥𝑣(𝑥) for all 𝑥∈Ω.

So we conclude that 𝑢_𝜖≥𝑢_𝛼>0 and 𝑣_𝜖≥𝑣_𝛿>0 in Ω.

Taking limits on both members of (2.16) and (2.6) as 𝜖→0⁺, we deduce that 𝑢_𝜖⇀𝑢 and 𝑣_𝜖⇀𝑣, for some 𝑢,𝑣∈𝐻¹₀(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 −Δ_𝑚𝑢(𝑥)+𝜆_Ω𝑣^𝑝𝑦𝑑𝑦=𝑢^𝛼(𝑥)+𝑣^𝛽(𝑥),𝑥∈Ω,−Δ_𝑛𝑣(𝑥)+𝜆_Ω𝑢^𝑞𝑦𝑑𝑦=𝑢^𝛾(𝑥)+𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω, (2.21) where −Δ_𝑚𝑢=−div(|∇𝑢|^𝑚−2∇𝑢) and −Δ_𝑛𝑢=−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 𝑊₀^1,𝑚(Ω) and 𝑊₀^1,𝑛(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 −Δ_𝑚(𝑥)𝑢(𝑥)+𝜆_Ω𝑣^𝑝(𝑥)𝑦𝑑𝑦=𝑢^𝛼(𝑥)+v^𝛽(𝑥),𝑥∈Ω,−Δ_𝑛(𝑥)𝑣(𝑥)+𝜆_Ω𝑢^𝑞(𝑥)𝑦𝑑𝑦=𝑢^𝛾(𝑥)+𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω, (2.22) where −Δ_𝑚(𝑥)𝑢=−div|∇𝑢|^{𝑚(𝑥)−2}∇𝑢,−Δ_𝑛(𝑥)𝑢=−div|∇𝑢|^{𝑛(𝑥)−2}∇𝑢 (2.23) are, respectively, the 𝑚(𝑥)-𝐿𝑎𝑝𝑙𝑎𝑐𝑖𝑎𝑛 and 𝑛(𝑥)-𝐿𝑎𝑝𝑙𝑎𝑐𝑖𝑎𝑛 and 𝑝(𝑥), 𝛼(𝑥), 𝛽(𝑥), 𝑛(𝑥), 𝑞(𝑥), 𝛾(𝑥), 𝛿(𝑥) are continuous functions on Ω satisfying suitable conditions. In this case, we have to work in the generalized Lebesgue-Sobolev spaces 𝑊₀^1,𝑚(𝑥)(Ω) and 𝑊₀^1,𝑛(𝑥)(Ω). See, for example, [21, 22] and the references therein, for more detailed information on this subject.

3. 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−Δ𝑢(𝑥)+𝜆_Ω𝑣^𝑝𝑦1𝑑𝑦=𝑢^𝛼(𝑥)(𝑥),𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^𝑞𝑦𝑑𝑦=𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω (3.1) 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−Δ𝑢(𝑥)+𝜆_Ω𝑣^𝑝𝑦1𝑑𝑦=𝑢^𝛼(𝑥)+|∇𝑢(𝑥)|^𝛾(𝑥),𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^𝑞𝑦𝑑𝑦=𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω, (3.2) or−Δ𝑢(𝑥)+𝜆_Ω𝑣^𝑝𝑦1𝑑𝑦=𝑢^𝛼(𝑥)+|∇𝑢(𝑥)|^𝛾(𝑥),𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^𝑞𝑦1𝑑𝑦=𝑣^𝛿(𝑥)𝑢(𝑥)(𝑥)𝑢(𝑥)(𝑥),𝑥∈Ω,,𝑣>0,𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω. (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−Δ𝑢(𝑥)+𝜆_Ω𝑣^+𝑝𝑦1𝑑𝑦=((𝑢⁺(𝑥))+𝜖)^𝛼(𝑥),𝑥∈Ω,−Δ𝑣+𝜆_Ω𝑢^+𝑞𝑦𝑑𝑦=𝑣⁺(𝑥)^𝛿𝑢(𝑥)(𝑥),𝑥∈Ω,=𝑣=0,𝑥∈𝜕Ω, (3.4) with 𝜖>0, to obtain approximate solutions (𝑢_𝜖,𝑣_𝜖). In this way, we avoid the singular term.

We have the following result.

Theorem 3.1. Let 0<𝛼,𝛿<1 be real numbers and 𝜆<0 a real parameter. Then, problem (3.1) possesses a positive solution (𝑢,𝑣)∈(𝐶²(Ω)∩𝐶(Ω))×(𝐶²(Ω)∩𝐶(Ω)).

Proof. Reasoning as in the proof of Theorem 2.1 we obtain, for each 𝜖>0, a solution (𝑢_𝜖,𝑣_𝜖) of problem (3.4). So, since 𝜆<0, we obtain −Δ𝑢_𝜖(𝑥)≥1𝑢⁺_𝜖(𝑥)+𝜖^𝛼>0,𝑥∈Ω,−Δ𝑣_𝜖(𝑥)≥𝑣⁺_𝜖(𝑥)^𝛿𝑢≥0,𝑥∈Ω,_𝜖(𝑥)=𝑣_𝜖(𝑥)=0,𝑥∈𝜕Ω. (3.5)
In view of the maximum principle, we obtain that 𝑢_𝜖>0 in Ω and 𝑣_𝜖≥0 in Ω. Consequently, −Δ𝑢_𝜖(𝑥)+𝜆_Ω𝑣^𝑝_𝜖𝑦1𝑑𝑦=𝑢_𝜖(𝑥)+𝜖^𝛼,𝑥∈Ω,−Δ𝑣_𝜖(𝑥)+𝜆_Ω𝑢^𝑞_𝜖𝑦𝑑𝑦=𝑣^𝛿_𝜖(𝑥)𝑢,𝑥∈Ω,_𝜖(𝑥)=𝑣_𝜖(𝑥)=0,𝑥∈𝜕Ω. (3.6)
If 𝑣_𝜖≡0, we obtain, in view of (3.6), that ∫_Ω𝑢^𝑞_𝜖(𝑦)𝑑𝑦=0, which contradicts the fact that 𝑢_𝜖>0 in Ω. Thus, 𝑣_𝜖≢0 and, because 𝑣_𝜖≥0, the maximum principle gives 𝑣_𝜖>0 in Ω.
The solution of problem (3.1) will be obtained by studying the limit when 𝜖→0. Thus, we may suppose that 0<𝜖<1. In view of this, we obtain from (3.6) that −Δ𝑢_𝜖(𝑥)>1𝑢_𝜖(𝑥)+1^𝛼𝑢,𝑥∈Ω,_𝜖(𝑥)=0,𝑥∈𝜕Ω. (3.7)
Let 𝜔_𝜖 be the unique positive solution of −Δ𝜔_𝜖(𝑥)=1𝑢_𝜖(𝑥)+1^𝛼𝜔,𝑥∈Ω,_𝜖(𝑥)=0,𝑥∈𝜕Ω. (3.8)
From the maximum principle, we deduce that 𝑢_𝜖>𝜔_𝜖>0 in Ω. Since 1/(𝑢_𝜖+1)^𝛼 is bounded, by using interior elliptic regularity, we obtain that there is 𝜔∈𝐶²(Ω) such that 𝜔_𝜖→𝜔 in 𝐶²(Ω′), for all Ω′⊂⊂Ω.
Using the Galerkin method and reasoning as in the proof of Theorem 2.1, we deduce that the approximate solutions 𝑢_𝜖,𝑣_𝜖 are uniformly bounded in 𝐻¹₀(Ω) with respect to 0<𝜖<1. In consequence, we conclude that 𝑢_𝜖⇀𝑢 and 𝑣_𝜖⇀𝑣 in 𝐻¹₀(Ω) in the weak sense. Hence, in view of (3.8), we obtain (𝑥)=1−Δ𝜔(𝑢(𝑥)+1)^𝛼𝜔(𝑥),𝑥∈Ω,=0,𝑥∈𝜕Ω. (3.9)
Invoking again the maximum principle, we get 𝜔>0 in Ω. Since 𝑢_𝜖>𝜔_𝜖 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.

4. 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−Δ𝑢(𝑥)+𝜆_Ω𝑢^𝑝𝑦𝑑𝑦=𝑢^𝛼(𝑥)(𝑥)𝑢(𝑥)𝑢(𝑥)+𝑓,𝑥∈Ω,>0,𝑥∈Ω,=0,𝑥∈𝜕Ω, (4.1) where 1<𝑝,𝛼<(𝑁+2)/(𝑁−2),𝑁≥3 and 𝑓∈𝐿²(Ω),𝑓≥0,𝑓≢0. Such a function 𝑓 is introduced in order to ensure that the solution is nonnegative and nontrivial. Note that 𝑢≡0 is a solution of−Δ𝑢(𝑥)+𝜆_Ω𝑢^𝑝𝑦𝑑𝑦=𝑢^𝛼(𝑥)𝑢(𝑥),𝑥∈Ω,=0,𝑥∈𝜕Ω, (4.2) and we do not have at our disposal the results in [19, 20].

Theorem 4.1. Suppose that 1<𝑝,𝛼<(𝑁+2)/(𝑁−2),𝑁≥3 and 𝜆<0. Then, for each 𝑓∈𝐿²(Ω), with ‖𝑓‖₂ sufficiently small, problem (4.1) possesses a nonnegative and nontrivial solution.

Proof. First of all, let us consider the auxiliary problem −Δ𝑢(𝑥)+𝜆_Ω𝑢^+𝑝𝑦𝑑𝑦=𝑢⁺(𝑥)^𝛼(𝑥)𝑢(𝑥)+𝑓,𝑥∈Ω,=0,𝑥∈𝜕Ω. (4.3)
As before, let us consider 𝐹∶ℝ^𝑚→ℝ^𝑚,𝐹=(𝐹₁,…,𝐹_𝑚), defined, for all 𝑖=1,…,𝑚, as 𝐹_𝑖(𝜉)=_Ω∇𝑢∇𝜑_𝑖+𝜆_Ω𝑢^+𝑝_Ω𝜑_𝑖−_Ω𝑢^+𝛼𝜑_𝑖−_Ω𝑓𝜑_𝑖, (4.4) where we are using the previous identifications.
For all 𝑖=1,…,𝑚, the following equations hold 𝐹_𝑖(𝜉)⋅𝜉_𝑖=_Ω∇𝑢⋅∇𝜉_𝑖𝜑_𝑖+𝜆_Ω𝑢^+𝑝_Ω𝜉_𝑖𝜑_𝑖−_Ω𝑢^+𝛼𝜉_𝑖𝜑_𝑖−_Ω𝑓𝜉_𝑖𝜑_𝑖, (4.5) and so 𝐹(𝜉),𝜉=_Ω|∇𝑢|²+𝜆_Ω𝑢^+𝑝_Ω𝑢−_Ω𝑢^+𝛼𝑢−_Ω𝑓𝑢. (4.6)
Using the classical immersions, we get 𝐹(𝜉),𝜉≥‖𝑢‖²+𝜆𝐶‖𝑢‖^𝑝+1−𝐶‖𝑢‖^𝛼−𝐶𝑓₂‖𝑢‖ (4.7) for all 𝑢∈𝕍_𝑚.
We now consider the function 𝑔(𝑡)=𝑡²+𝜆𝐶𝑡^𝑝+1−𝐶𝑡^𝛼+1,𝑡≥0, (4.8) and note that 𝑔(𝑡)=𝑡²1+𝜆𝐶𝑡^𝑝−1−𝐶𝑡^𝛼−1. (4.9)
So, we may find a 𝑡₀>0, sufficiently small, such that 𝑔𝑡₀=𝑡²₀1+𝜆𝐶𝑡₀^𝑝−1−𝐶𝑡₀^𝛼−1>0. (4.10)
As consequence, we can choose ‖𝑓‖₂, small enough, such that 𝑔𝑡₀=𝑡²₀1+𝜆𝐶𝑡₀^𝑝−1−𝐶𝑡₀^𝛼−1>𝐶𝑓₂𝑡₀, (4.11) and, taking 𝑢∈𝕍_𝑚 such that ‖𝑢‖=𝑡₀, we get 𝐹(𝜉),𝜉>0,𝜉=‖𝑢‖=𝑡₀,𝑢∈𝕍_𝑚, (4.12) and note that such a 𝑡₀ does not depend on 𝑚.
Proceeding as in the proof of Theorem 2.1, we obtain, for each 𝑚∈ℕ, an approximate solution 𝑢_𝑚∈𝕍_𝑚 satisfying ‖𝑢_m‖≤𝑡₀ and, for all 𝑖=1,…,𝑚, _Ω∇𝑢_𝑚∇𝜑_𝑖+𝜆_Ω𝑢⁺_𝑚^𝑝_Ω𝜑_𝑖−_Ω𝑢⁺_𝑚^𝛼𝜑_𝑖−_Ω𝑓𝜑_𝑖=0. (4.13)
Thus, for all 𝜑∈𝕍_𝑚, we have _Ω∇𝑢_𝑚∇𝜑+𝜆_Ω𝑢⁺_𝑚^𝑝_Ω𝜑−_Ω𝑢⁺_𝑚^𝛼𝜑−_Ω𝑓𝜑=0. (4.14)
Since we are working in the superlinear and subcritical case and using the previous arguments, we obtain that there is a function 𝑢∈𝐻¹₀(Ω) satisfying _Ω∇𝑢∇𝜑+𝜆_Ω𝑢^+𝑝_Ω𝜑−_Ω𝑢^+𝛼𝜑−_Ω𝑓𝜑=0 (4.15) for all 𝜑∈𝐻¹₀(Ω).
We now use the weak maximum principle to conclude that 𝑢≥0 in Ω. From the fact that 𝑓≢0, the solution is not the trivial one. Then, 𝑢 is a solution of our original problem (4.1).

Remark 4.2. Note that, when we are working with a problem with variational structure the function 𝑓 may be discarded because we may use the Mountain Pass Theorem. In this case the critical level given by this theorem is positive and so we may conclude that the solution obtained in this way is not null. However, in our case, we can not dispose of the variational techniques. In view of this and to show that we obtain a nontrivial solution, we had to introduce the function 𝑓.

Acknowledgments

This work was done while the second author was working as a Visiting Professor in the Postgraduate Program in Mathematics of the Universidade Federal do Pará, Brazil. He would like to express his profound gratitude to Professor Giovany M. Figueiredo for his warm hospitality. A. Cabada was partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, Project MTM2010-15314. F. J. S. A. Corrêa was partially supported by CNPq, Grant 301603/2007-3.