On the Uniqueness of the Limiting Solution to a Strongly Coupled Singularly Perturbed Elliptic System

This article is concerned with a strongly coupled elliptic system modeling the steady state of two or more populations that compete in some regions. We prove the uniqueness of the limiting configuration as the competing rate tends to infinity, under suitable conditions. The proof relies on properties of limiting solution and Maximum principle.


Introduction
In this paper, we consider the following strongly coupled system of elliptic equations: where i u denotes the density of the i-th population, While if 0 ij β > for some , i j , the system becomes strongly coupled. System (1.1) (or its parabolic case) has been investigated by many workers [1]- [6], and various existing results have been developed. In particular, when 2 m = , Lou and Ni [2] characterized the existence of nonconstant positive solutions both for the small and large competition cases, while those in [4] [5] were concerned with the existence of positive solutions in relation to a pair of curves in the ( ) 1 2 , a a -plane for both large and small cross-diffusion cases. For the existing results concerning the case when 3 m ≥ , we refer to [6] and references therein.
According to Gause's principle of competitive exclusion, two competing species cannot coexist under strong competition. The migration or the spatial distribution changes the situation and all the species survive but have disjoint habits, which is called spatial segregation [7]. To investigate such a phenomenon, we will focus on the so called strong competition regime, that is when the parameter k diverges to +∞ , while the positive coefficients ij b remain fixed.
In the classic Lotka-Volterra competition model (1.2), it is proved that k- it is proved that the free boundary consists of two parts: a regular set, which is a 1, C α locally smooth hypersurface, and a singular set of Hausdorff dimension not greater than 2 n − ; furthermore, in dimension 2, then free boundary consists in a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, see for example [8] [9] [14]. Unlike the symmetric case, the asymmetric case (i.e. when ij for some , i j ) shows the emergence of spiraling nodal curves, still meeting at locally isolated points with finite vanishing order [15].
A further related problem is the study of the uniqueness and least energy property of the limiting configuration as k → +∞ . In the case of three species and in dimension 2, Conti et al. [16] proved the uniqueness and least energy properties for the limiting state. That is, the solution of system (1.2) (when 0 i i a b = = ) converges, as k → +∞ , to the minimizer of a variational problem.
In [13], Wang and Zhang generalized the result to arbitrary dimensions and arbitrary number of species. In [17], Arakelyan and Bozorgnia also proved the uniqueness of the limiting solution to system (2). On the other hand, coming back to the strongly coupled system, Zhou et al. [18] [19] study the asymptotic behavior of solutions to system (1.1). They obtained the similar spatial segregation results and established the uniform C α ( 0 1 α < < ) bounds for solutions to system (1.1).
In this paper, we continue the study of system (1.1), we are concerned with the uniqueness of the limiting configuration of system (1.1). In order to simplify the notations, throughout the paper we assume 1 We only consider nonnegative solutions, that is, those 0 k i u ≥ in its domain for all i.
Our result is as follows. We note that Theorem 1.1 has already been proved in [19], where the uniqueness, also the least energy properties for the limiting state has been established. Their method originally stated in [13], is based on computing the derivative of the energy functional with respect to the geodesic homotopy between u and a comparison to an energy minimizing map v with same boundary values.
Our proof is different from the one in [13] [19]. In fact, our method follows the mainstream of [17], based on the properties of limiting solutions and Maximum principle. Compared with the work of [19], we in fact give a new proof of the uniqueness of the limiting configuration. Our proof doesn't require regular results of the free boundary. So in this sense, our proof is straightforward and simple.
Note that the study of strong-competition limits in corresponding elliptic or parabolic system is of interest not only for questions of spatial segregation in population, as here and in [20] [21], but also is key to the understanding of phase separation of Gross Pitaevskii systems of modeling Bose-Einstein condensates, see [22]- [27] and reference therein. Furthermore, the study on other aspects of segregation triggered by strong competition, starting from two pioneer-  [33] for long-range interaction models. The rest of the paper is organized as follows: In section 2, we introduce a transformation and recall some preliminary results, which are essential to the proof of the main results. In Section 3, we prove the uniqueness of the system (1.1) in the limiting case as k tends to infinity.

Some Preliminary Results
In this section, we mention some known results for the solutions of system (1.1), which play an important role in our study. To begin with, for every index i, we Then the Jacobian determinant Now we recall some estimates and compactness properties of solutions to system (1.1).
be defined as in (2.1). Then k z is a nonnegative solution of (2.2), and for every 0 such that for all 2) if we define for each index i: Remark 2.1. By (2.4) and Theorem 8.2 in [14], we have that each element of ( ) is actually global Lipschitz continuous on Ω .

Uniqueness of the Limiting Configuration
In this section, we prove Theorem 1.1. We perform a change of variable in order to deal with the problem in a different setting. Let By the definition of ( ) In this setting, we consider the corresponding singular space , , : and 0 for .
We claim that: for all 1 i m ≤ ≤ , and we complete the proof of Lemma 3.1.  In view of Lemma 3.1 we define the following quantities belong to  and satisfying (3.4). We set P and Q as defined above. If 0 P > is attained for some index 1 i m ≤ ≤ , then we have 0 P Q = > . Moveover, there exist another index 0 0 j i ≠ and a point 0 x ∈ Ω , such that: Proof Let the maximum 0 P > be attained for the 0 th i component. According to the previous lemma, we know that This contradiction implies that 0 Q > . By analogous proof, one can see that if P be non-positive then Q will be non-positive as well. Next, assume the maximum P is attained at a point  .
The same argument shows that Q P ≤ which yields P Q = . Hence, we can  are belong to the class  and satisfy (3.4). For them, we set P and Q as above. Then, we consider two cases 0 P ≤ and 0 P > . If we assume that 0 P ≤ then Lemma 3.2 implies that 0 Q ≤ . This leads to for every 1 i m ≤ ≤ , and x ∈ Ω . This provides that Now, suppose 0 P > , we show that this case leads to a contradiction. Let the value P is attained for some 0 i , then due to Lemma 3.2 there exist 0 x ∈ Ω and 0 0 j i ≠ such that: Let Γ be a fixed curve starting at 0 x and ending on the boundary of Ω . Since Ω is connected, then one can always choose such a curve belonging to Ω . By the disjointness and smoothness of On the other hand, in view of Lemma 3.2 we have which implies that P is attained at the interior point , then clearly the only possibility is .
Following the lines of the proof of Lemma 3.2, we find some 0 It is easy to see that there exists a ball ( ) 1 1 r B x (without loss of generality one keeps the same notation) In view of the maximum principle and above steps we obtain: Then we take ( ) such that 1 x stands between the points 0 x and 2 x along the given curve Γ . According to the previous arguments for the point 2 x we will find an index

Conclusions and Further Works
The study of the asymptotic behavior of singular perturbed equations and sys- tems of elliptic or parabolic type is very broad and subject of research. In this paper, we study a strongly coupled elliptic system arising in competing models in population dynamics. We give an alternative proof of the uniqueness of the limiting configuration as k → +∞ under suitable conditions. We remark that the approach here is different from the one in [19]. Our proof doesn't require regular results of the free boundary. So in this sense, our proof is straightforward and simple.
Finally, we mention that there are many interesting problems for further study.
Note that we prove the uniqueness of the limiting solutions to a strongly coupled elliptic system, naturally to ask whether this result can be extended to the corresponding parabolic system? Up to our knowledge, the uniform Hölder bounds for parabolic setting is unknown, and both the asymptotics and the qualitative properties of the limit segregated profiles remain a challenge, this will be the object of a forthcoming paper.

Founding
The work is partially supported by PRC grant NSFC 11601224.