The estimates on the energy functional of an elliptic system with Neumann boundary conditions
© Zeng; licensee Springer 2013
Received: 19 June 2013
Accepted: 13 August 2013
Published: 28 August 2013
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.
Keywordselliptic system estimates energy functional
where is a small parameter, Ω is a bounded domain in (), . f and g are nonlinearities having superlinear and subcritical growth at infinity.
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.
(S3) , , for , , where , .
where , , , .
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).
Assume that is continuous and satisfies the following structure assumptions.
(f1) for and near . as , for some if , and if .
(f2) There exists a constant such that for , in which .
(f3) The function is strictly increasing.
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).
(2.2) is a functional defined over the Hilbert space .
It can be observed that the following orthogonal splitting holds: , here, , . We set , .
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
and , .
Lemma 3.1 Supposing the assumptions in Theorem 2.3 hold, admits the following properties:
(i)1 , , ,
(ii)1 , ,
Proof Proof of (i)1. By the definition of , , then , where u and v are the solutions of (1.1), that follows .
by (S2), we have , , so .
So, we get the properties (i)1.
For p is a number between 2 and , , we deduce . It contradicts with the original assumption of .
Thus, we obtain . □
Proposition 3.2 (Theorem 1.1 in )
Under assumptions (H), there exists such that for any , problem (1.1) has nonconstant positive solutions . Moreover, both functions and attain 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).
We claim that the function , defined in (3.9), has the following properties:
(i)2 , , ;
(ii)2 For any , if , then ;
(iii)2 , as .
Following from assumption (S2), we obtain .
Combined with (3.4), . By Lemma 3.1, , so .
then by (3.12), . We proved the property (ii)2.
The proof of (iii)2 is similar to (ii)1. Then we complete the proof of the claim.
In fact, (3.13) and (3.14) is a contradiction to the nature (ii)2 of , that is, for any , if , the value of must be smaller than 0.
Having reached a contradiction, this completes the proof of Theorem 2.3. □
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).
- Gierer A, Meinhardt H: A theory of biological pattern formation. Kybernetik 1972, 12: 30-39. 10.1007/BF00289234View ArticleGoogle Scholar
- Lin CS, Ni WM, Takagi I: Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 1988, 72: 1-27. 10.1016/0022-0396(88)90147-7MathSciNetView ArticleGoogle Scholar
- Ni WM, Takagi I: On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type. Trans. Amer. Math. Soc. 1986, 297: 351-368. 10.1090/S0002-9947-1986-0849484-2MathSciNetView ArticleGoogle Scholar
- Ni WM, Takagi I: On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math. 1991, 44: 819-851. 10.1002/cpa.3160440705MathSciNetView ArticleGoogle Scholar
- Ni WM, Takagi I: Locating peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 1993, 70: 247-281. 10.1215/S0012-7094-93-07004-4MathSciNetView ArticleGoogle Scholar
- Del Pino M, Felmer P: Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana Univ. Math. J. 1999, 48: 883-898.MathSciNetView ArticleGoogle Scholar
- Ni WM, Wei J: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 1995, 48: 731-768. 10.1002/cpa.3160480704MathSciNetView ArticleGoogle Scholar
- Wang ZQ: On the existence of multiple single-peaked solution for a semilinear Neumann problem. Arch. Ration. Mech. Anal. 1992, 120: 375-399. 10.1007/BF00380322View ArticleGoogle Scholar
- Grossi M, Pistoia A, Wei J: Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. Calc. Var. Partial Differential Equations 2000, 11: 143-175. 10.1007/PL00009907MathSciNetView ArticleGoogle Scholar
- Grossi M, Pistoia A: On the effect of critical points of distance function in superlinear elliptic problems. Adv. Differential Equ 2000, 5: 1397-1420.MathSciNetGoogle Scholar
- Li YY, Nirenberg L: The Dirichlet problem for singularly perturbed elliptic equations. Comm. Pure Appl. Math. 1998, 51: 1445-1490. 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-ZMathSciNetView ArticleGoogle Scholar
- Wei J: On the boundary spike layer solutions to a singularly perturbed Neumann problem. J. Differential Equations 1997, 134: 104-133. 10.1006/jdeq.1996.3218MathSciNetView ArticleGoogle Scholar
- Wei J: On the interior spike layer solutions of a singularly perturbed semilinear Neumann problem. Tohoku Math. J. 1998, 50: 159-178. 10.2748/tmj/1178224971MathSciNetView ArticleGoogle Scholar
- Wei J: On the interior spike solutions for some singular perturbation problems. Proc. Roy. Soc. Edinb. Sect A 1998, 128: 849-874. 10.1017/S030821050002182XView ArticleGoogle Scholar
- Avila AI, Yang J: On the existence and shape of least energy solutions for some elliptic systems. J. Differential Equations 2003, 191: 348-376. 10.1016/S0022-0396(03)00017-2MathSciNetView ArticleGoogle Scholar
- Busca J, Sirakov B: Symmetry results for semilinear elliptic systems in the whole space. J. Differential Equations 2000, 163: 41-56. 10.1006/jdeq.1999.3701MathSciNetView ArticleGoogle Scholar
- Clement P, De Figueiredo DG, Mitidieri E: Positive solutions of semilinear elliptic systems. Comm. Partial Differential Equations 1992, 17: 923-940. 10.1080/03605309208820869MathSciNetView ArticleGoogle Scholar
- Pistoia A, Ramos M: Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Differential Equations 2004, 201: 160-176. 10.1016/j.jde.2004.02.003MathSciNetView ArticleGoogle Scholar
- Pistoia A, Ramos M: Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions. Nonlinear Differ. Equ. Appl. 2008, 15: 1-23. 10.1007/s00030-007-4066-8MathSciNetView ArticleGoogle Scholar
- Gidas B, Ni WM, Nirenberg L:Symmetry of positive solutions of non-linear elliptic equations in . Adv. in Math. Suppl. Stud. 7. Mathematical Analysis and Applications. Part A 1981, 369-402.Google Scholar
- Abbondandolo A, Felmer P, Molina J: An estimate on the relative Morse index for strongly indefinite functionals. Electron. J. Differ. Equ. Conf. 2001, 6: 1-11.MathSciNetGoogle Scholar
- Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.