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Existence of solutions for a class of singular elliptic systems with convection term

We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines pseudomonotone operator theory, sub and supersolution method and perturbation arguments involving singular terms.


Introduction
In this work, we focus our attention on the existence of solutions for the following class of elliptic system with convection term where Ω is a bounded domain with smooth boundary and g i : R 2N → [0, +∞), i = 1, 2, are positive continuous functions belong to L ∞ (R 2N ). We consider the system (S) ± in a singular case assuming α i , β i , ∈ [0, 1) for i = 1, 2.
Hereafter (u, v) is a solution to (S) ± if u, v ∈ C 2 (Ω) ∩ H 1 0 (Ω) are both positive in Ω and satisfy the equations of (S) ± in the classical sense.
Nonlinear singular boundary value problems are mathematically challenging and important for applications.
They arise in several physical situations such as fluid mechanics, pseudoplastics flow, chemical heterogeneous catalysts, non-Newtonian fluids, biological pattern formation, for more details about this subject, we cite the papers of Fulks & Maybe [17], Callegari & Nashman [6,7] and the references therein.
Systems (S) ± can be see as a version of the singular scalar equations in Ω, u > 0 in Ω, with α, β > 0 and g : R N → R be a continuous function verifying some technical conditions. Several works are devoted to classes of problems covering (P ) ± . For instance, see the papers of Aranda [5], Ghergu & Radulescu [19,20], Giarrusso & Porru [18], Lair & Wood [22], Zhang [26] and references therein. Problem (P ) ± without a convection term, that is g = 0 was also investigated.  [24]. From the above commentaries, we observe that in recent years singular elliptic problems with convection term has received few attention. Motivated by this fact, our aim is to show the existence of solutions for a class of elliptic systems where the nonlinearity besides a singular term has a convection term. The proof combines results involving pseudomonotone operators, sub and supersolution method and perturbation arguments involving singular terms. We emphasize that our study complete those made in [1], [2] and [24], in the sense that in those papers the authors did not consider the case where the nonlinearity has a convection term, and also [4], because a different type of singular term was considered. The method used in the present work is different from those applied in the aforementioned papers.
Our main result is the following: Theorem 1.1 Assume that g i : R N → R are continuous functions and α i , β i ∈ [0, 1) for i = 1, 2. Then, the systems (S) ± has a solution.
The proof of Theorem 1.1 is done in Sections 3 and 4. The first main technical difficulty is that the nonlinearities of (S) ± depend of gradient of the solution, which is more a complicating factor. Indeed, for the scalar case, an interesting result is proved by Kazdan & Kramer [21] and Leon [23], where the authors develop a sub and supersolution method for scalar problem where the nonlinearity depends of the gradient. Instead, the counter-part of this result for systems with gradient terms is not known in the literature. Thus, we do not know a result involving sub and supersolution that could be use to establishes the existence of solution for this class of system. To overcome this difficulty, we show in Section 2 a result, see Theorem 2.1, which can be see as a sub and supersolution method for systems whose nonlinearity depends of gradient.
The second main difficulty in the proof of Theorem 1.1 is associated with the fact that the sub and supersolution method in its version involving maximum principle cannot be used directly for systems involving the gradient of the solution. Moreover, the way as the singularities appear in the system (S) ± is a difficult point to work with maximum principle. In order to overtake the stated problem we first introduce a parameter ε > 0 in (S) ± , giving rise to regularized systems for (S) ± whose study is relevant for our initial problem. Then, for the regularized systems, we combine variational methods with the sub-supersolution one to prove the existence of a solution . This solution (u ε , v ε ) is located in some rectangle formed by the sub and supersolution, independent for ε > 0, which does not contains zero for all ε > 0. Then, a positive solution of (S) ± is obtained by passing to the limit as ε → 0. This is based on a priori estimates and Hardy-Sobolev inequality. The positivity of the solution is derived from the independence of the subsolution of the regularized systems on ε.
The rest of this article is organized as follows: In section 2 we state and prove a general theorem about sub and supersolution method for systems with convection term. Sections 3 and 4 contain the proof of Theorem 1.1.

An auxiliary result
The main goal in this section is to prove the Theorem 2.1 below, which is a key point in the proof of Theorem 1.1. An interesting point related to Theorem 2.1 is the fact that it is a result of sub and supersolution whose the proof is made using pseudomonotone operator theory.
Theorem 2.1 Let H, G : Ω×R + ×R + ×R N ×R N → R continuous functions function verifying the following conditions: Given T, S > 0, there exist C > 0 and α, β ∈ (0, 1), such that Assume that Proof. Here, we will adapt some arguments found in Leon [23]. Firstly, we introduce two new functions Moreover, for each l ∈ (0, 1), we consider Using the above functions, we will work with the ensuing auxiliary system Setting the functions and we define the operator B : E → E ′ given by with being the usual norm in H 1 0 (Ω). Using the hypotheses on H and G together with the definition of H 1 , H 2 , G 1 , G 2 , γ 1 and γ 2 , we can prove the ensuing properties for operator B:

I) B is continuous:
The proof of this property follows by using the fact that H 1 , G 1 belong to L ∞ .

II) B is bounded:
Here, the boundedness of B is understood in the sense that if U ⊂ E is a bounded set, then B(U) ⊂ E ′ is also bounded. This property also follows using the boundedness of H 1 and G 1 .

III) B is coercive:
Here, it is enough to prove that Using again the boundedness of H 1 and G 1 , we derive showing that

IV) B is pseudomonotone:
First of all, we recall that B is a pseudomonotone operator if (u n , v n ) ⇀ (u, v) in E and verifies lim sup (2.2) In our case, the weak limit (u Thereby, the above limits combined with (2.1) load to The properties I) − IV ) allow us to use [16,Theorem 3.3.6] to conclude that B is surjective. Therefore, there exists (u, v) ∈ E such that implying that (u, v) is a solution of (S 1 ). Now, our goal is showing that We will show only (i), because the same arguments can be used to prove (ii).
or equivalently, showing that (u − u) + = 0, from where it follows that u ≤ u. To prove that u ≤ u, we choose (φ, ψ) = ((u − u) + , 0) as a test function. Repeating the above arguments, we get Since (u, v) is a subsolution, it follows that or equivalently, showing that (u−u) + = 0, and so, u ≤ u. Combining (2.3) with the definition of H 2 and G 2 , it follows that and showing that (u, v) is a solution for system (AS).

Existence of solution for system (S) −
In this section, we will study the existence of solution for the following singular elliptic system where Ω is a bounded domain in R N with smooth boundary, g i : R 2N → [0, +∞) are positive continuous functions belong to L ∞ (R 2N ) and α i , β i ∈ [0, 1). Our approach consists in considering for ǫ > 0 the approximated system For this class of system it is possible to find sub and supersolution which do not depend of ǫ. For example, by using the positive eigenfunction associated with the first eigenvalue, we can find easily a subsolution (u, u). To get the supersolution, we observe that any large constant M > 0 can be used to get a supersolution (u, v) = (M, M) with M > u ∞ . Now, let us consider the functions and G ǫ (x, s, t, η, ξ) = 1 which are well defined in Ω × R + × R + × R N × R N .

Existence of solution for system (S) +
In this section, we will study the existence of solution for the following singular elliptic system (S) +                    −∆u = 1 v α 1 + v β 1 + g 1 (∇u, ∇v) in Ω, −∆v = 1 u α 2 + u β 2 + g 2 (∇u, ∇v) in Ω u, v > 0 in Ω, u = v = 0 on ∂Ω, by supposing the same hypotheses of Section 3. The existence of solution for (S) + can be obtained by using the same arguments explored in the previous section. The unique difference is in the construction of the supersolution that we will work. Here, the idea is the following: Fix R > 0 large enough such that Ω ⊂ B R (0) and denote by e the unique solution of the problem