A system of superlinear elliptic equations in a cylinder

The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Ω=Ω′×(0,a)⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\Omega '\times (0,a)\subset {\mathbb {R}}^n$$\end{document}, where Ω′⊂Rn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega '\subset {\mathbb {R}}^{n-1}$$\end{document} is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} with another superlinear (or linear) equation defined at the bottom of the cylinder Ω′×{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega '\times \{0\}$$\end{document}. Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.


Introduction
For scalar equations of the form proved by a priori estimates for the scalar Eq. (1.1) have been extended to corresponding elliptic nonlinear coupled systems [4,5,12,13,15]. For instance, the following superlinear system where Ω ⊂ R n is a bounded domain and p, q > 1, is usually referred to as the coupled Lane-Emden system and has been widely investigated in the last few years (see [10,16,24] and the references therein). Such problems arise in the study of multicomponent reaction diffusion processes and in the modeling of several physical phenomena such as pattern formation and population evolution (see [25] and the references therein). The solutions in most of the cases represent densities and thus positive solutions of the systems are of particular interest. The exponents (p, q) in system (1.2) interplay, compensating each other, which play a crucial role in the questions of existence and nonexistence of positive solutions.
In [4], Clément, de Figueiredo and Mitidieri used a method which was developed in [11] for the case of one equation to obtain L ∞ a priori bounds. For another coupled system studied by de Figueiredo and Yang [15], the difficulties of obtaining a priori bounds were due to the presence of gradients in the nonlinear terms. The authors had to use some norm with weights depending on the distance to the boundary of the domain. They obtained a priori bounds via the so called blow-up method which was introduced by Gidas-Spruck [17] for the scalar case. In [5], the authors found L ∞ a priori bounds with different exponent assumptions imposed on the nonlinear terms; the technique used in their work is based on the work of Brezis and Turner [3] for one equation, in which they combined the Hardy-Sobolev inequality with interpolation techniques. In [5] the Brezis-Turner exponent assumption was replaced by conditions that involve two curves in the (p, q) plane. We remark that the method introduced by Brezie-Turner was the first general way to obtain uniform bounds of positive solutions and has become a classical way. Many other problems like reaction-diffusion systems and Ambrosetti-Prodi type problems have been solved by this method (see [14,21]).
The objective of this paper is to study the existence of positive solutions of a particular semi-linear elliptic system defined in a cylinder Ω = Ω ×(0, a) ⊂ R n , where Ω ⊂ R n−1 is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Ω with another superlinear (or linear) equation defined at the bottom Ω ×{0} of the cylinder. Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). Extending the method of Brezis-Turner [3] to this kind of system, we provide a priori L ∞ bounds for all positive solutions when the nonlinear terms satisfy certain growth conditions. The approach we use consists of using the Hardy-Sobolev inequality and a suitable fixed point theorem. Unlike the setting in [3] where the nonlinear term f (x, u) is defined on Ω×R, in our framework f is non-local and we have to distinguish two cases, depending on the space dimension. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis-Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.

The Main Result
In this paper we consider a system of equations on a cylindrical domain Ω = The particularity of this system is that it couples two unknowns u(x) and v(x ) which are defined on different domains. We can think of Ω as a jar or a cylindrical habitat containing two interacting substances or species: the substance u(x) (say a gas, insects, birds...) is distributed in the interior of the jar or habitat Ω, while the substance v(x ) (say a fluid, plants, worms...) is located at the bottom Ω × {0} of the jar or on the ground of the habitat. A simple model of such a time independent interacting system is . , x n−1 ), ν denotes the exterior normal to the boundary ∂Ω, and γ, η are exponents with γ > 1 and η ≥ 1.
Here, we assume that the vertically cumulated effect of the substance u(x), x ∈ Ω, interacts with the substance v(x ) at the bottom Ω , hence the term a 0 u η (x , x n ) dx n in the second equation; on the other hand, the substance v(x ) at the bottom Ω interacts with the substance u(x) via a continuous coefficient function h : Ω → R + , which we may consider decreasing with increasing height x n .
The operator Δ (n−1) with Dirichlet boundary condition in the second equation is invertible, and we can insert the expression into the first equation of the system, to obtain the non-local equation Our aim is to prove the following result: Then Eq. (2.2), and hence system (2.1), has a positive solution u ∈ W 2,q (Ω), 1 ≤ q < ∞. Remark 2.1. Notice that for n = 3, 4, we are always in case 1), since then .
The proof follows the ideas of the influential paper by Brezis-Turner [3], in which a single equation with a super-linear non-linearity was considered. It is interesting to note that the maximal exponent in the article of Brezis-Turner was n+1 n−1 . For η = 1, the maximal exponent for γ is 2n+2 n , 3 ≤ n ≤ 6, n 2 −1 n 2 −4n+1 , n ≥ 7 which is larger than n+1 n−1 , this is due to the regularizing effect of the inverted operator (−Δ (n−1) ) −1 .
We have not seen such type of coupled systems in the literature. Of course, one can consider many different versions of such couplings.

L p regularity on the cylinder
The proof of Theorem 2.1 depends on a priori estimates of the solutions and a related existence theorem. The L p theory presented here is to pave the way to get the a priori bound. In this part we will concentrate on showing that a weak solution of the equation will also be a strong solution which is twice weakly differentiable. The proof of the regularity is based on the a priori estimates below. In view of the mixed boundary conditions and the special shape of the domain, we will do an even reflection on the bottom of the cylinder to reduce the problem to a familiar case for which we can refer to the ninth chapter in [18].

L p a priori estimate
We define the space H 1 cyl (Ω) as the closure in H 1 (Ω) of the set C 1 where C = C(n, p, Ω i , Ω).
Since the interior estimate does not require the boundary condition, the proof of this lemma follows from the same proof of Theorem 9.11 [18].
Estimate on the bottom and the top: Proof. We extend u and f to Ω ×(−a, a) by even reflection, that is, by setting for x n < 0. It follows that the extended functions, sayũ andf , satisfy the same equation of (3.1) weakly in Ω × (−a, a). To prove this we take an arbitrary test function ϕ ∈ C 1 cyl (Ω × (−a, a)), then since u is a weak solution of (3.1) on Ω, we have As ϕ ∈ C 1 in Ω × (0, a) and ϕ = 0 on ∂Ω , we can take φ = ϕ in Ω × (0, a), then On the other hand, due to the even reflection, from (3.3), we get Consequently, we have , a)).
Estimate on the side: where C = C(n, p, Ω s , Ω).
Proof. In the proof of Lemma 3.1.2, we extended u and f to Ω ×(−a, a) by even reflection, and we proved that the extended functionũ is a weak solution of (3.1) in Ω ×(−a, a) with f replaced byf . In this case, each point x 0 ∈ ∂Ω ×{0} is a boundary point of Ω × (−a, a) on the side, we then can proceed as in the proof of Lemma 3. 0,a)) . We therefore derive Combining all the estimates above, we get the following result.
Global L p estimate and regularity: Proof. (see a similar proof of Theorem 2.2.3 [26]) From the boundary estimate we conclude that for x 0 ∈ ∂Ω, there exists a neighborhood U (x 0 ) such that According to Heine-Borel theorem, there exists a finite open covering Using the theorem on the partition of unity, we can choose functions η 0 , η 1 , . . . , η N such that In the next lemma we eliminate the dependence of u on the right.
Since v N satisfies (3.12) weakly, then On the other hand, since and (3.13), we see In the following we prove v = 0. Indeed, multiplying with v on both sides of Eq. (3.16), we get Ω |∇v| 2 dx = 0, so ∇v = 0, combining with the boundary condition then

Regularity:
With the above a priori estimate we can get the following existence result: Proof. The existence of the strong solution follows as in Th.9.15 [18]. Here we present the main points of the proof. We start from the L 2 regularity.  (3.17) the constant C depending only on V , Ω. The proof of L 2 interior regularity is the same as Theorem 1 ([8] section 6.3.1). In order to get the boundary regularity, we extend u and f to Ω ×(−a, a) as we did in Lemma 3.1.2. The extended functionũ andf satisfy the same equation of (2.1) weakly in Ω × (−a, a). Since the bottom Ω × {0} is inside of Ω × (−a, a) after the extension, then the proof of regularity near the bottom Ω × {0} is the same as L 2 interior regularity. Considering u = 0 on ∂Ω , then the regularity near the side of the cylinder is the same as Theorem 4 ( [8] section 6.3.2). Thus we have: is a weak solution of (3.1), then u ∈ H 2 (Ω), and we have the estimate We are now in a position to prove Lemma 3.2.1 with 2 < p < ∞. In fact, given that we have the same L p a priori estimates as in chapter 9 [18], the interior regularity result follows directly from Lemma 9.16 [18]. After we did the even reflection, the case of local boundary regularity is handled similarly as the Lemma 9.16 as well.

A Priori Bounds
In this section we will see that the growth conditions imposed on the nonlinear terms play an important role in acquiring a priori bounds for all positive solutions of system (2.1). These terms are embedded into different L p spaces as the dimension n varies. Depending on the size of η (the growth of nonlinearity in u) we will find two different growth restrictions for γ (the growth of the nonlinearity in v).
First, we state the following Hardy-type estimate in H 1 cyl (Ω), which is the preparatory step of the technical aspects of the proof. The constant C may change from line to line, we will use C for a generic constant.
dx dx n ; we start the proof from the inner integral. Consider u n ∈ C ∞ cyl (Ω); for fixed x n , u n (x , x n ) is a function of x , then by Hardy's inequality [22] we have and then integrating along the x n direction, (4.1) as n → ∞. This implies that {u n } is a Cauchy sequence in H 1 cyl (Ω), then there exist n such that for n, m ≥ n , is a Cauchy sequence in L 2 (Ω) and hence In fact, . Then we complete the proof by letting n → ∞ in (4.1).
The next lemma is a variant of the Hardy-inequality.
Proof. By the Hölder inequality, Applying Lemma 4.1 and Sobolev's embedding theorem to the respective term in (4.2) we obtain u δ τ

Then (4.3) becomes the desired inequality.
In what follows we let J 1 denote the first positive eigenfunction satisfying where λ 1 is the first eigenvalue of −Δ (n−1) and J 1 is normalized so that Remark 4.1. It is known that J 1 (x ) > 0 in Ω and it follows from Hopf's Lemma that The basic a priori bound we prove is the following.
We first prove some lemmas.
Proof. Since u ∈ H 1 cyl (Ω) is a weak solution of (4.4), we have {0, a}). The left side of the equation yields, using that u| ∂Ω ×[0,a] = 0 and ∂ ν J 1 | Ω ×{0,a} = 0 Since by assumption h(x) has the positive lower bound h m , then where k > 0 will be chosen below. Since we consider non-negative solutions, then a 0 u η (x , x n ) dx n is non-negative, and by the maximum principle, (−Δ (n−1) ) −1 a 0 u η (x , x n ) dx n is non-negative. Therefore Next, choose k such that h m · a · k γ−1 ≥ (λ 1 ) 2 + 1, thus which implies t is bounded, and also Ω u(x)J 1 dx < C.
is bounded, and using Remark 4.1 we obtain, This completes the proof of Lemma 4.3.
Next, we show a Poincaré type inequality in W 1,p cyl (Ω).
From (4.8), (4.15), Lemma 4.3 and the Hölder inequality we now deduce Again applying the Hardy inequality with τ = α we get where Then as before, we take Now that ρ ≤ 2 * η , from (4.7), (4.19), (4.20) and (4.21) we get We can then conclude from (4.22) that Now combining (4.21) with γ r ≤ ρ ✩ and γ < β n , we are going to find a best r to have the largest γ. So first we take ρ = 2 * η . Thus Since β n is increasing with respect to r and the largest γ is decreasing with respect to r, we can let ), (4.23) and thus, from (4.21) Like the first case, the choice of ρ in (4.16) is possible for the second case of Theorem 2.1. Based on the above two cases (1 ≤ η < 4n (n−1)(n−2) , η ≥ 4n (n−1)(n−2) ), the proof of Lemma 4.5 is complete.
Next, suppose that η ≤ η ≤ 2 * , it then follows that 2 ≤ n r . Then we will get an improved uniform L p bound of f (x, u) by showing an improved uniform L p bound of u. To see this we first consider Similarly, in the case 2 > (n − 1)η r * , we can replace ( r * η ) ✩ by +∞. In the case . (4.26) From (4.25), we deduce ≤ C. Noting that (4.27) where the last inequality follows by elementary calculations, using (4.23). So we see that f (·, u) is bounded in an improved L p space, if 2 ≤ (n−1)η r * . Then taking p = ( r * η ) ✩ γ , by (3.10) and the Sobolev inequality, we have, . From (4.27), we see ( r * η ) ✩ γ * > r * , which means we get a better uniform L p bound of u. Afterwards, we repeat the computation of (4.26) and get Iterating (4.26)-(4.28), finally, we will derive Thus, we have completed the proof of Theorem 4.1.

Fixed Point Theorem and Existence of the Solution
In this section we complete the proof of Theorem 2.1. We first show a maximum principle for the Poisson equation with mixed boundary conditions.
where ν is the outer unit normal vector to ∂Ω. Then the solution of (5.1) satisfies: Proof. If the claim were not true, then there exists a x 0 ∈ Ω such that u(x 0 ) < 0. Without loss of generality, we suppose u(x 0 ) = min x∈Ω u(x) < 0.
We are now in the position to complete the proof of Theorem 2.1.
Proof of Theorem 2.1. For every fixed u ≥ 0 in C 1 (Ω), by the Lax-Milgram theorem we know there exists a unique solution for Eq. (2.2), which we denote by w u . That is, To solve problem (2.2), we define the mapping u → w u =: F (u). If there is a fixed point of F in C 1 (Ω) such that F (u) = u, we are done. Now we check that F satisfies the following fixed point theorem ( [9], Theorem 3.1; [19], Theorem 1).
F : C 1 (Ω) → C 1 (Ω) a compact mapping, acting in the cone of nonnegative functions, will have a fixed point Step 1: F : C 1 (Ω) → C 1 (Ω) is a compact mapping. It is easy to see that F is continuous, since it is a composition of continuous maps. Then, let A ⊂ C 1 (Ω) be a bounded set, for u ∈ A we have 2) thus f (x, u) ∈ L ∞ (Ω) and {f (x, u), u ∈ A} is uniformly bounded. Since −Δ (n) w u = f (x, u), by Lemmas 3.1.6 and 3.2.1, w u ∈ W 2,q (Ω), q large enough, and lies in a bounded set in W 2,q (Ω). Then by Morrey's inequality, we get for q > n, w u ∈ C 1,γ (Ω), that is where γ = 1 − n q . Therefore we have for every x, y inΩ, and ∀u ∈ A Hence, ∀ > 0, we take δ = ( C ) γ /1 then, if |x − y| < δ, {w u } satisfies |Dw u (x) − Dw u (y)| ≤ C|x − y| γ < which means {w u , u ∈ A} is uniformly bounded and equicontinuous in C 1 (Ω). According to the Arzelà-Ascoli theorem, it is in a compact set in C 1 (Ω). Hence, F is a compact mapping from C 1 (Ω) to C 1 (Ω).
Step 2: F maps the non-negative cone in C 1 (Ω) into itself. For this we are going to prove that when u is fixed non-negative, then w u is non-negative. Indeed, w u satisfies where f (x, u) = f (x) = h(x)[(−Δ (n−1) ) −1 a 0 u η (x , x n )dx n ] γ . By (5.2) f ∈ L ∞ (Ω) so that f ∈ L p (Ω) for any p > 1 when u is fixed in C 1 (Ω).
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