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.
Chen, L. and Jüngel, A. (2004) Analysis of a Multi-Dimensional Parabolic Population Model with Strong Cross-Diffusion. SIAM Journal on Mathematical Analysis, 36, 301-322. https://doi.org/10.1137/S0036141003427798
Lou, Y. and Ni, W. (1996) Diffusion, Self-Diffusion, and Cross-Diffusion. Journal of Differential Equations, 131, 79-13. https://doi.org/10.1006/jdeq.1996.0157
Lou, Y., Ni, W. and Wu, Y. (1998) On the Global Existence of a Cross-Diffusion System. Discrete and Continuous Dynamical Systems, 4, 193-203. https://doi.org/10.3934/dcds.1998.4.193
Ruan, W.H. (1996) Positive Steady-State Solutions of a Competing Reaction-Diffusion System with Large Cross Diffusion Coefficients. Journal of Mathematical Analysis and Applications, 197, 558-578. https://doi.org/10.1006/jmaa.1996.0039
Ruan, W.H. (1999) A Competing Reaction-Diffusion System with Small Cross-Diffusions. Canadian Applied Mathematics Quarterly, 7, 69-91.
Pao, C.V. (2005) Strongly Coupled Elliptic Systems and Applications to Lotka-Volterra Models with Cross-Diffusion. Nonlinear Analysis, 60, 1197-1217. https://doi.org/10.1016/j.na.2004.10.008
Gause, G.F. (1932) Experimental Studies on the Struggle for Existence: 1. Mixed Population of Two Species of Yeast. Journal of Experimental Biology, 9, 389-402. https://doi.org/10.1242/jeb.9.4.389
Conti, M., Terracini, S. and Verzini, G. (2005) Asymptotic Estimates for the Spatial Segregation of Competitive Systems. Advances in Mathematics, 195, 524-560. https://doi.org/10.1016/j.aim.2004.08.006
Caffarelli, L.A., Karakhanyan, A.L. and Lin, F. (2009) The Geometry of Solutions to a Segregation Problem for Nondivergence Systems. Journal of Fixed Point Theory and Applications, 5, 319-351. https://doi.org/10.1007/s11784-009-0110-0
Conti, M., Terracini, S. and Verzini, G. (2005) A Variational Problem for the Spatial Segregation of Reaction-Diffusion Systems. Indiana University Mathematics Journal, 54, 779-815. https://doi.org/10.1512/iumj.2005.54.2506
Dancer, E.N., Wang, K. and Zhang, Z. (2012) Dynamics of Strongly Competing Systems with Many Species. Transactions of the American Mathematical Society, 364, 961-1005. https://doi.org/10.1090/S0002-9947-2011-05488-7
Dancer, E.N. and Zhang, Z. (2002) Dynamics of Lotka-Volterra Competition Systems with Large Interaction. Journal of Differential Equations, 182, 470-489. https://doi.org/10.1006/jdeq.2001.4102
Wang, K. and Zhang, Z. (2010) Some New Results in Competing Systems with Many Species. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 27, 739-761. https://doi.org/10.1016/j.anihpc.2009.11.004
Tavares, H. and Terracini, S. (2012) Regularity of the Nodal Set of the Segregated Critical Configuration under a Weak Reflection Law. Calculus of Variations and Partial Differential Equations, 45, 273-317. https://doi.org/10.1007/s00526-011-0458-z
Terracini, S., Verzini, G. and Zilio, A. (2019) Spiraling Asymptotic Profiles of Competition-Diffusion Systems. Communications on Pure and Applied Mathematics, 72, 2578-2620. https://doi.org/10.1002/cpa.21823
Conti, M., Terracini, S. and Verzini, G. (2006) Uniqueness and Least Energy Property for Strongly Competing Systems. Interfaces and Free Boundaries, 8, 437-446. https://doi.org/10.4171/IFB/150
Arakelyan, A. and Bozorgnia, F. (2017) Uniqueness of Limiting Solution to a Strongly Competing System. Electronic Journal of Differential Equations, 96, 1-8.
Zhou, L., Zhang, S., Liu, Z. (2012) Uniform Hölder Bounds for a Strongly Coupled Elliptic System with Strong Competition. Nonlinear Analysis, 75, 6210-6219. https://doi.org/10.1016/j.na.2012.06.017
Zhang, S., Zhou, L., Liu, Z.H. (2017) Uniqueness and Least Energy Property for Solutions to a Strongly Coupled Elliptic System. Acta Mathematica Sinica, English Series, 33, 419-438. https://doi.org/10.1007/s10114-016-5686-x
Dancer, E.N. and Du, Y. (1995) Positive Solutions for a Three-Species Competition System with Diffusion. I. General Existence Results. Nonlinear Analysis, 24, 337-357. https://doi.org/10.1016/0362-546X(94)E0063-M
Dancer, E.N. and Du, Y. (1995) Positive Solutions for a Three-Species Competition System with Diffusion. II. The Case of Equal Birth Rates. Nonlinear Analysis, 24, 359-373. https://doi.org/10.1016/0362-546X(94)E0064-N
Caffarelli, L.A. and Lin, F. (2008) Singularly Perturbed Elliptic Systems and Multi-Valued Harmonic Functions with Free Boundaries. American Mathematical Society, 21, 847-862. https://doi.org/10.1090/S0894-0347-08-00593-6
Dancer, E.N., Wang, K. and Zhang, Z. (2012) The Limit Equation for the Gross-Pitaevskii Equations and S. Terracini’s Conjecture. Journal of Functional Analysis, 262, 1087-1131. https://doi.org/10.1016/j.jfa.2011.10.013
Noris, B., Tavares, H., Terracini, S. and Verzini, G. (2010) Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition. Communications on Pure and Applied Mathematics, 63, 267-302. https://doi.org/10.1002/cpa.20309
Soave, N. and Zilio, A. (2015) Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case. Archive for Rational Mechanics and Analysis, 218, 647-697. https://doi.org/10.1007/s00205-015-0867-9
Wei, J. and Weth, T. (2008) Asymptotic Behaviour of Solutions of Planar Elliptic Systems with Strong Competition. Nonlinearity, 21, 305-317. https://doi.org/10.1088/0951-7715/21/2/006
Zhang, S. and Liu, Z. (2015) Singularities of the Nodal Set of Segregated Configurations. Calculus of Variations and Partial Differential Equations, 54, 2017-2037. https://doi.org/10.1007/s00526-015-0854-x
De Silva, D. and Terracini, S. (2018) Segregated Configurations Involving the Square Root of Laplacian and Their Free Boundaries. Calculus of Variations and Partial Differential Equations, 58, Article No. 87. https://doi.org/10.1007/s00526-019-1529-9
Terracini, S., Verzini, G. and Zilio, A. (2016) Uniform Hölder Bounds for Strongly Competing Systems Involving the Square Root of the Laplacian. Journal of the European Mathematical Society, 18, 2865-2924. https://doi.org/10.4171/JEMS/656
Terracini, S., Verzini, G. and Zilio, A. (2014) Uniform Hölder Regularity with Small Exponent in Competing Fractional Diffusion Systems. Discrete and Continuous Dynamical Systems, 34, 2669-2691. https://doi.org/10.3934/dcds.2014.34.2669
Verzini, G. and Zilio, A. (2014) Strong Competition versus Fractional Diffusion: The Case of Lotka-Volterra Interaction. Communications in Partial Differential Equations, 39, 2284-2313. https://doi.org/10.1080/03605302.2014.890627
Caffarelli, L., Patrizi, S. and Quitalo, V. (2017) On a Long Range Segregation Model. Journal of the European Mathematical Society, 19, 3575-3628. https://doi.org/10.4171/JEMS/747
Soave, N., Tavares, H., Terracini, S. and Zilio, A. (2018) Variational Problems with Long-Range Interaction. Archive for Rational Mechanics and Analysis, 228, 743-772. https://doi.org/10.1007/s00205-017-1204-2