Positive Solutions for Some Nonlinear Elliptic Systems in Exterior Domains of R2

and Applied Analysis 3 asymptotic behavior of the unique solution of the associated homogeneous system, where we have denoted by ω : aHDφ ch, θ : bHDψ dh, 1.3 and the functions HDφ and h are the harmonic functions defined, respectively, by 1.4 and 1.5 below. The system of two equations in 1.1 has been treated in exterior domains of R, n ≥ 3 in 19 , and existence of positive bounded continuous solutions is established. The main difficulty in the present work is the case of the domain D ⊂ R2. More precisely, the function h defined by 1.5 behaves as ln |x| at infinity for n 2, unlike the case n ≥ 3 where this function is bounded at infinity. Throughout this paper, we denote by HDφ the unique bounded continuous solution of the Dirichlet problem Δw 0 in D, w/∂D φ, lim |x|→ ∞ w x h x 0, 1.4 where φ is a nonnegative continuous function on ∂D. The function h is defined on D by h x 2π lim |y|→ ∞ GD ( x, y ) . 1.5 First, we recall the following result about this function h. Proposition 1.1 see 16 . The function h defined by 1.5 is harmonic and positive in D and satisfies lim x→ z∈∂D h x 0, lim |x|→ ∞ h x ln|x| 1. 1.6 Taking into account these notations, we use some potential theory tools and an approximating sequence in order to prove the following first result concerning the existence of a unique positive continuous solution to the boundary value problem: Δu p x u , in D u/∂D aφ lim x→∞ u x ln|x| c, 1.7 where γ ≥ 1, φ is a nontrivial nonnegative continuous function on ∂D and a, c are two nonnegative constants with a c > 0. More precisely we establish the following. 4 Abstract and Applied Analysis Theorem 1.2. Let p be a nonnegative function such that the function p̃ γpωγ−1 belongs to the Kato class K D . Then problem 1.7 has a unique positive continuous solution satisfying for each x ∈ D c0ω x ≤ u x ≤ ω x , 1.8 where ω is defined in 1.3 and the constant c0 ∈ 0, 1 . Next we exploit this result to prove the existence of a positive continuous solution u, v to the system 1.1 . For this aim we denote by ω0 : a ch, 1.9 θ0 : b dh, 1.10 and we need to assume the following hypothesis on the functions p and q. H p and q are nonnegative measurable functions in D such that x −→ p x θ 0 x ωα−1 0 x , x −→ q x ω 0 x θ β−1 0 x 1.11 are in K D . Using the Schauder fixed point, we prove the following main result. Theorem 1.3. Under the hypothesis H , the problem 1.1 has a positive continuous solution u, v satisfying for each x in D c1ω x ≤ u x ≤ ω x , c2θ x ≤ v x ≤ θ x , 1.12 where ω, θ are defined by 1.3 and c1, c2 ∈ 0, 1 . In order to state these results and for the sake of completeness, we give in the sequel some notations, and we recall some properties of the Kato class K D studied in 16, 17 . Let us denote by B D the set of Borel measurable functions inD and by B D the set of nonnegative ones. We denote also by C0 D the set of continuous functions in D having limit zero at ∂D, by C D ∪ {∞} {f ∈ C D : lim|x|→∞f x exists} and by C0 D {f ∈ C D : lim|x|→∞f x 0}. We note that C D ∪ {∞} is a Banach space endowed with the uniform norm ‖f‖∞ supx∈D|f x |. First we recall that if φ is a nonnegative continuous function on ∂D, then from 20, page 427 the functionHDφ ∈ C D ∪ {∞} and satisfies limx→∞HDφ x C > 0. For any f in B D , we denote by Vf the Green potential of f defined on D by


Introduction
The study of nonlinear elliptic systems has a strong motivation, and important research efforts have been made recently for these systems aiming to apply the results of existence and asymptotic behavior of positive solutions in applied fields. Coupled nonlinear Shrödinger systems arise in the description of several physical phenomena such as the propagation of pulses in birefringent optical fibers and Kerr-like photorefractive media, see 1, 2 . Stationary elliptic systems arise also in other physical models like non-Newtonian fluids: pseudoplastic fluids and dilatant fluids 3, 4 , non-Newtonian filtration 5 , and the turbulent flow of a gas in porous medium 6,7 . They also describe other various nonlinear phenomena such as chemical reactions, pattern formation, population evolution where, for example, u and v represent the concentrations of two species in the process. As a consequence, positive solutions of such are of interest.
For some recent results on the qualitative analysis and the applications of positive solutions of nonlinear elliptic systems in both bounded and unbounded domains we refer to [8][9][10][11][12][13][14][15] and the references therein.

Abstract and Applied Analysis
In these works various existence results of positive bounded solutions or positive blowing-up ones called also large solutions have been established, and a precise global behavior is given. We note also that several methods have been used to treat these nonlinear systems such as sub-and super-solutions method, variational method, and topological methods.
In this paper, we consider an unbounded domain D in R 2 with a nonempty compact boundary ∂D consisting of finitely many Jordan curves and noncontaining zero. We fix two nontrivial nonnegative continuous functions ϕ and ψ on ∂D and some nonnegative constants, a, b, c, d such that a c > 0 and b d > 0, and we will deal with the existence of a positive continuous solution in the sense of distributions to the system: where α ≥ 1, β ≥ 1, r ≥ 0, s ≥ 0 and p, q are two nonnegative functions satisfying some hypotheses related to the Kato class K D defined and studied in 16, 17 by means of the Green function G x, y of the Dirichlet Laplacian in D.
Our method is based on some potential theory tools which we apply to give an existence result for equations by an approximation argument, then we use the result for equations to prove, by means of the Schauder fixed point theorem, the existence result for the system 1.1 .
As far as we know, there are no results that contain existence of positive solutions to the elliptic system 1.1 in the case where α > 0 and β > 0 and the weights p x and q x are singular functions.
The study of 1.1 is motivated by the existence results obtained in 18 to the following system where λ, μ are nonnegative constants, the functions f, g : 0, ∞ → 0, ∞ are nondecreasing and continuous. More precisely, it was shown in 18 that if the functions p : pf θ and q : qg ω belong to the Kato class K D , then there exist λ 0 > 0 and μ 0 > 0 such that for each λ ∈ 0, λ 0 and μ ∈ 0, μ 0 , the system 1.
Taking into account these notations, we use some potential theory tools and an approximating sequence in order to prove the following first result concerning the existence of a unique positive continuous solution to the boundary value problem: where γ ≥ 1, ϕ is a nontrivial nonnegative continuous function on ∂D and a, c are two nonnegative constants with a c > 0. More precisely we establish the following.
where ω is defined in 1.3 and the constant c 0 ∈ 0, 1 .
Next we exploit this result to prove the existence of a positive continuous solution u, v to the system 1.1 . For this aim we denote by ω 0 : a ch, 1.9 and we need to assume the following hypothesis on the functions p and q.
H p and q are nonnegative measurable functions in D such that Using the Schauder fixed point, we prove the following main result.
In order to state these results and for the sake of completeness, we give in the sequel some notations, and we recall some properties of the Kato class K D studied in 16, 17 . Let us denote by B D the set of Borel measurable functions in D and by B D the set of nonnegative ones. We denote also by C 0 D the set of continuous functions in D having limit zero at ∂D, by and we recall that if f ∈ L 1 loc D and V f ∈ L 1 loc D , then we have in the distributional sense see 21, page 52 Let X t , t > 0 be the Brownian motion in R 2 and P x be a probability measure on the Brownian continuous paths starting at x. For any function q ∈ B D , we define the kernel V q by where E x is the expectation on P x and τ D inf{t > 0 : X t / ∈ D}. If q is a nonnegative function in D such that V q < ∞, the kernel V q satisfies the following resolvent equation see 21, 22 So for each u ∈ B D such that V q|u| < ∞, we have where ρ x min 1, δ x and δ x denotes the Euclidian distance from x to the boundary ∂D of D.
This Kato class is rich enough as it can be seen in the following example. Remark 1.6. Let p > 1 and λ, μ ∈ R such that λ < 2 − 2/p < μ. Then using the Hölder inequality and the same arguments as in the proof of the precedent example it follows that for each f ∈ L p D , the function defined in D by f x / 1 |x| μ−λ δ x λ belongs to K D .
Next, we recall some properties of K D . ii The function iv For any nonnegative superharmonic function v in D and all x ∈ D, one has D G x, y v y q y dy ≤ α q v x · 1.21 The following compactness results will be used and they are proved, respectively, in 17 and 16 .  As a consequence of these Propositions, we obtain the following.

Corollary 1.10. Let q be a nonnegative function in K D . Then the family of functions:
x −→ 1 ω 0 x D G x, y ω 0 y p y dy; p ≤ q 1.24 Proof. Since then the result follows from Propositions 1.8 and 1.9.
The following result will play an important role in the proofs of Theorems 1.2 and 1.3.

Proof of Theorem 1.2
First we give two Lemmas that will be used for uniqueness. Proof. Let u be a nonnegative continuous solution of 1.7 . First, we will prove that u This implies that the function v ε ω εh − u satisfies

8 Abstract and Applied Analysis
Hence by 20, page 465 , we get u x ≤ ω x εh x in D. Since ε is arbitrary, this implies that u x ≤ ω x for each x ∈ D. Now, since p γpω γ−1 ∈ K D , then pu γ−1 ∈ K D . Hence it follows from Propositions 1.8 and 1.9 that V aH D pu γ−1 and V chpu γ−1 belong to C D with boundary value zero, which implies that V pωu γ−1 belongs to C D with boundary value zero. So, V pu γ belongs to C D with boundary value zero. Consequently, using Corollary 7 page 294 in 20 , we deduce that the function u − ω V pu γ is a classical harmonic in D with boundary value zero and satisfying lim |x| → ∞ u x − ω x V pu γ x / ln |x| 0. Thus by 20, page 419 , we have u − ω V pu γ 0 in D. So u ω − V pu γ and this proves necessity. Now, we prove sufficiency. Let u be a nonnegative continuous function in D satisfying the integral equation u ω − V pu γ . Since p is nonnegative and p γpω γ−1 ∈ K D , then u ≤ ω and pu γ−1 ∈ K D . This implies, by using Propositions 1.8 and 1.9 that V aH D ϕpu γ−1 and V chpu γ−1 are in C D with boundary value zero. Consequently, V pu γ is in C D with boundary value zero. Hence, Δu Δω − Δ V pu γ pu γ in the sense of distributions and u is a solution of 1.7 .

Now we prove Theorem 1.2
Proof of Theorem 1.2. First we show that problem 1.7 has at most one continuous solution. Let u, v be two continuous solutions of 1.7 . Then, by Lemma 2.
Then we have ζ ≥ 0 and z V pζz 0 in D. Using Lemma 2.1, we deduce that z 0 and so u v.
Next, we prove the existence of a positive continuous solution to 1.7 . We recall that ω aH D ϕ ch and p γpω γ−1 ∈ K D . Put c 0 e −α p where the constant α p is defined in Proposition 1.7. We define the nonempty closed bounded convex set Λ by Let T be the operator defined on Λ by We will prove that T maps Λ to itself. Indeed, for each u ∈ Λ, we have
Next, we prove that T is nondecreasing on Λ. Let u 1 , u 2 ∈ Λ such that u 1 ≤ u 2 . Since for each y ∈ D, the function t → p y t − p y t γ is nondecreasing on 0, ω y we deduce that Now, we consider the sequence u k k defined by u 0 ω−V p pω and u k 1 Tu k . Clearly u 0 ∈ Λ and u 1 Tu 0 ≥ u 0 . Thus, using the fact that Λ is invariant under T and the monotonicity of T , we deduce that Hence, the sequence u k k converges to a measurable function u ∈ Λ. Therefore, by applying the monotone convergence theorem, we deduce that u satisfies the following equation: Applying the operator I V p. on both sides of 2.10 , we deduce by using 1.16 and 1.17 that u ω − V pu γ .

2.11
Now, let us verify that u is a solution of the problem 1.7 . Since p γpω γ−1 ∈ K D , then by Proposition 1.7, we have p ∈ L 1 loc D . Now, using the following inequality: Therefore u ∈ C D . Now, using Propositions 1.1 and 1.9, we deduce that lim x → ∂D V ph x 0. In addition, since H D ϕ is bounded in D, we deduce from Proposition 1.7 lim x → ∂D V p x 0. So that lim x → ∂D V pω x 0. This in turn implies that lim x → ∂D V pu γ 0. Which together with 2.11 imply that u /∂D aϕ. On the other hand, we have Using Propositions 1.9, 1.8, and 1.1, we deduce that 1/ ln |x| V pu γ x tends to zero as |x| → ∞ and consequently lim |x| → ∞ u x / ln |x| c. This implies that u is a positive continuous solution of 1.7 . This completes the proof of Theorem 1.2.

Remark 2.3.
Let p 0 γ max 1, ϕ ∞ pω γ−1 0 , where ω 0 is given by 1.9 . Then we have 0 ≤ p ≤ p 0 . So if we assume that p 0 ∈ K D , then p ∈ K D and α p ≤ α p 0 . Moreover, the solution u of 1.7 satisfies also the inequality: 2.16 Next we give the proof of Theorem 1.3.
In order to use a fixed point theorem, we consider the nonempty closed convex set Γ defined by For λ, χ ∈ Γ, we consider the following system: Δy p θ − θ 0 χ r y α , in D Δz q x ω − ω 0 λ s z β , in D y /∂D aϕ, z /∂D bψ, 3.2