We consider an elliptic system of the form , in Ω with Neumann boundary conditions, where Ω is a domain in , f and g are nonlinearities having superlinear and subcritical growth at infinity. We prove the existence of nonconstant positive solutions of the system, and estimate the energy functional on a configuration space by a different technique, which is an important step in the proof of the solution’s concentrative property. We conclude that the least energy solutions of the system concentrate at the point of boundary, which maximizes the mean curvature of ∂Ω.
Keywords:elliptic system; estimates; energy functional
We are concerned with the following singularly perturbed system with Neumann conditions:
Problem (1.1) arises in many applied models concerning biological pattern formations. For example, the steady states in the Keller-Segel model, the Gierer-Meinhardt model, see [1,2] for more details. Problem (1.1) has been studied extensively for last twenty years. The motivation for the study of such a problem goes back to the pioneering work of [2,3] concerning the scalar case (single equation),
They proved a priori estimates, existence of least energy solutions and the concentrative properties of the solution. Furthermore, in [4,5], Ni and Takagi proved the existence of a nontrivial solution to problem (1.2) for ε small enough. They showed that attains its maximum value at a point , and the subsequences of converge to P, which is the maximum point of mean curvature on ∂Ω.
The subject was studied by many authors for both Neumann and Dirichlet boundary conditions. There are many well-known results about (1.2). Del Pino and Felmer in  introduced shorter and more elementary arguments with respect to those in [4,7]. Wang in  obtained multiple solutions of (1.2) by using Ljusternik-Schnirelman method. In , Grossi et al., obtained a solution of (1.2) with k maxima points, k is a given positive integer. We refer the reader to [10-14] for further references.
As far as we know, Avial and Yang  were the first to approach the singularly perturbed system (1.1) with Neumann boundary conditions; they considered (1.1) with special nonlinearities , (). By means of a dual variational formulation, they proved that there exist nontrivial positive solutions and in , which have global maximum point at different points.
A more direct approach was proposed in [16-18]. In these papers, the authors extend the idea, which is introduced by Del Pino and Felmer in , to system (1.1). In , Pistoia and Ramos proved the least energy solutions of system (1.1) concentrate at a point of the boundary, which maximizes the mean curvature of the boundary of Ω. Pistoia and Ramos  consider system (1.1) with Dirichlet boundary condition, they proved the existence of the least energy solutions. The solutions are concentrated, as ε goes to zero, at a point of Ω, which is maximized in distance to the boundary of Ω.
Besides, it is known that the underlying minimax theorem associated to ground-state level of (1.1) is an infinite-dimensional linking, this is in contrast with (1.2). We refer the reader to [18,21] for more details on this.
In this paper, we prove the existence of nonconstant positive solutions of system (1.1), and estimate the energy functional of (1.1) on the configuration space (defined in Section 2) by a different technique, which is compared with . This estimation is an important step in the proof of , where denotes the mean curvature of ∂Ω at the boundary point P. We conclude the least energy solutions of system (1.1), concentrated at the point of boundary, which maximizes the mean curvature of the boundary of Ω.
2 Statement of main results
The assumption to is a typical superlinear subcritical one, as in , we assume that the following holds.
Remark 2.1 Examples of nonlinearities satisfying (S1)-(S3) are
We should point out that (S1)-(S3) are the natural extension of the assumptions for the scalar case (single equation). Let us recall the assumptions on single equation such as (1.2).
Remark 2.2 Assumption (S1) is the ‘system edition’ of (f1). (f2) is the famous Ambrosetti-Rabinowitz superlinear condition , which has appeared in most of studies for superlinear problems. In fact, it implies that the super-quadratic condition on . It has been used in a crucial way not only in establishing the mountain-pass geometry of the functional, but also in obtaining bounds of (PS) sequences. Assumption (S2) implies that , , which play a important roll in the proof of the existence of system’s solutions. So it is the ‘system edition’ of (f2).
equation (1.1) becomes
We define the norm
Remark 2.5 The estimation (2.3) is an important step in the proof of
where denotes the mean curvature of ∂Ω at the boundary point P. So we can conclude the least energy solutions of system (1.1) concentrate at a point of the boundary, which maximizes the mean curvature of the boundary of Ω.
3 Proof of Theorem 2.3
Again by (3.1),
So, we get the properties (i)1.
To prove (ii)1 is equal to show
By (S1), there exist C such that
As the same,
Now, we turn to the prove of (iii)1. By (3.1),
in the same way,
Proposition 3.2 (Theorem 1.1 in )
Under assumptions (H), there existssuch that for any, problem (1.1) has nonconstant positive solutions. Moreover, both functionsandattain their maximum value at some unique and common point. (The assumption (H) is composed of (S1), (S3) and the following (3.5).)
Remark 3.3 We will compare our assumptions (S1)-(S3) with the conditions (H) of Proposition 3.2 in the following proof of Theorem 2.3.
Proof of Theorem 2.3 The existence of solutions of (1.1) can follow the steps of Theorem 1.1 in . They use some ideas introduced by Del Pino and Felmer , and differ from the method of Ni and Takagi. It needs to be pointed out that (S2) implies the following conditions:
The assumption (H) in Proposition 3.2 is composed of (S1), (S3) and (3.5). By Proposition 3.2, the existence of solutions can be proved under (H). So, we can get the existence of solutions of (1.1) under (S1)-(S3).
Thus, by (3.8), we have
First, we prove (i)2. In fact, by (3.1) and (3.9),
Then by (3.10),
By (S3), we get
The proof of (iii)2 is similar to (ii)1. Then we complete the proof of the claim.
Having reached a contradiction, this completes the proof of Theorem 2.3. □
There are no financial competing interests in this manuscript. There are no non-financial competing interests (political, personal, religious, ideological, academic, intellectual, commercial or any other) to declare in relation to this manuscript.
There is only one author in the manuscript. JZ designed the study. She carried out the studies, and drafted the manuscript. The author read and approved the final manuscript.
The author is supported by the project of ‘Youth Innovation,’ funded by the Department of Science and Technology of Fujian province (2011J05003), and supported by the Projects A of the Educational Department of Fujian Province (JA11053).
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