- Research Article
- Open Access
Existence of Solutions for Elliptic Systems with Nonlocal Terms in One Dimension
© Alberto Cabada et al. 2011
- Received: 2 June 2010
- Accepted: 26 August 2010
- Published: 2 September 2010
We study the solvability of a system of second-order differential equations with Dirichlet boundary conditions and non-local terms depending upon a parameter. The main tools used are a dual variational method and the topological degree.
- Elliptic System
- Integral Term
- Radial Solution
- Topological Degree
- Nonlocal Term
where is a domain in , a survey was given by De Figueiredo in . The specific case of one-dimensional systems, motivated by the problem of finding radial solutions to an elliptic system on an annulus of , has been considered by Dunninger and Wang  and by Lee , who have obtained conditions under which such a system may possess multiple positive solutions.
On the other hand, systems of two equations that include non-local terms have also been considered recently. These are of importance because they appear in the applied sciences, for example, as models for ignition of a compressible gas, or general physical phenomena where temperature has a central role in triggering a reaction. In fact their interest ranges from physics and engineering to population dynamics. See for instance . The related parabolic problems are also of great interest in reaction-diffusion theory; see [5–7] where the approach to existence and blow-up for evolution systems with integral terms may be found.
where , and . First we consider (1.2) as a perturbation of the nonlocal system and prove that if and grow linearly, then (1.2) has a solution provided is not too large. Afterwards, assuming that and are monotone, we will give estimates on the growth of these functions in terms of the parameter to ensure solvability. This will be done on the basis of some spectral analysis for the linear part and a dual variational setting.
We first study the invertibility of the linear part of (1.2).
In view of the previous lemma we will assume
Let us introduce the basic assumption
Then problem (1.2) has a solution.
Thus we may invoke the properties of the Leray-Schauder degree (see, e.g., ) to deduce the existence of a solution for (3.3) with which is our problem (1.2).
one obtains that
Notice that and are the Fenchel transform of and (see ).
it follows that is a strictly monotone operator (see ). Hence, when (4.29) holds, has a unique critical point. The argument of Claim 2 in previous theorem shows that there is a one-to-one correspondence between critical points of and the solutions to (1.2). In consequence, the solution of problem (1.2) is unique.
The authors are indebted to the anonymous referees for useful hints to improve the presentation of the paper. The first and the second authors were partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2007-61724. The third author was supported by FCT, Financiamento Base 2009.
- de Figueiredo DG: Nonlinear elliptic systems. Anais da Academia Brasileira de Ciências 2000, 72(4):453-469. 10.1590/S0001-37652000000400002View ArticleMathSciNetGoogle Scholar
- Dunninger DR, Wang H: Multiplicity of positive radial solutions for an elliptic system on an annulus. Nonlinear Analysis: Theory, Methods & Applications 2000, 42(5):803-811. 10.1016/S0362-546X(99)00125-XView ArticleMathSciNetGoogle Scholar
- Lee Y-H: Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus. Journal of Differential Equations 2001, 174(2):420-441. 10.1006/jdeq.2000.3915View ArticleMathSciNetGoogle Scholar
- Corrêa FJSA, Lopes FPM: Positive solutions for a class of nonlocal elliptic systems. Communications on Applied Nonlinear Analysis 2007, 14(2):67-77.MathSciNetGoogle Scholar
- Deng W, Li Y, Xie C: Blow-up and global existence for a nonlocal degenerate parabolic system. Journal of Mathematical Analysis and Applications 2003, 277(1):199-217. 10.1016/S0022-247X(02)00533-4View ArticleMathSciNetGoogle Scholar
- Zhang R, Yang Z: Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system. Applied Mathematics and Computation 2008, 200(1):267-282. 10.1016/j.amc.2007.11.012View ArticleMathSciNetGoogle Scholar
- Li F, Chen Y, Xie C: Asymptotic behavior of solution for nonlocal reaction-diffusion system. Acta Mathematica Scientia. Series B 2003, 23(2):261-273.MathSciNetGoogle Scholar
- Brezis H: Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, France; 1983:xiv+234. Théorie et applicationsGoogle Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications. Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleGoogle Scholar
- Zeidler E: Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer, New York; 1986:xxi+897.View ArticleGoogle Scholar
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar
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