On a nonlocal eigenvalue problem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 30 (2001) no. 1, pp. 41-61. 
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     title = {On a nonlocal eigenvalue problem},
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Wei, Juncheng; Zhang, Liqun. On a nonlocal eigenvalue problem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 30 (2001) no. 1, pp. 41-61. http://archive.numdam.org/item/ASNSP_2001_4_30_1_41_0/

[1] A. Bose - G. Kriegsman, Stability of localized structures in non-local reaction-diffusion equations, Methods Appl. Anal. 5 (1998), 351-366. | MR | Zbl

[2] E.N. Dancer, On stability and Hopf bifurcations for chemotaxis systems, to appear in: "Proceedings of IMS Workshop on Reaction-Diffusion Systems 2000" . | MR

[3] M. Del Pino, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system, Indiana Univ. Math. J. 43 (1994), 703-728. | MR | Zbl

[4] M. Del Pino - P. Felmer - M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Asymp. Anal., to appear. | MR

[5] A. Doelman - A. Gardner - T.J. Kaper, Stability of singular patterns in the 1-D Gray-Scott model: A matched asymptotic approach, Physica D. 122 (1998), 1-36. | Zbl

[6] A. Doelman - A. Gardner - T.J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Technical Report, Center for BioDynamics, Boston University, submitted.

[7] A. Doelman - T. Kaper - P.A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity 10 (1997), 523-563. | MR | Zbl

[8] P. Freitas, Bifurcation and stability of stationary solutions on nonlocal scalar reaction diffusion equations, J. Dyn. Differential Equations 6 (1994), 613-629. | MR | Zbl

[9] P. Freitas, A non-local Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburg Sect. A 124 (1994), 169-188. | MR | Zbl

[10] P. Freitas, Stability of stationary solutions for a scalar nonlocal reaction diffusion equation, Quart. J. Mech. Appl. Math. 48 (1995), 557-582. | MR | Zbl

[11] A. Gierer - H. Meinhardt, A theory of biological pattern formation, Kybemetik (Berlin) 12 (1972), 30-39.

[12] P. Gray - S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci. 38 (1983), 29-43.

[13] P. Gray - S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B → 3B, B → C, Chem. Eng. Sci. 39 (1984), 1087-1097.

[14] J.K. Hale - L.A. Peletier - W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis, SIAM J. Appl. Math. 61 (2000), 102-130. | MR | Zbl

[15] D. Iron - M.J. Ward, A metastable spike solution for a non-local reaction-diffusion model, SIAM J. Appl. Math. 60 (2000), 778-802. | MR | Zbl

[16] D. Iron - M.J. Ward - J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D., to appear. | MR | Zbl

[17] C.-S. Lin - W.-M. Ni, "On the diffusion coefficient of a semilinear Neumann problem", Calculus of variations and partial differential equations (Trento, 1986), 160-174, Lecture Notes in Math., 1340, Springer, Berlin-New York, 1988. | MR | Zbl

[18] C.B. Muratov - V.V. Osipov, Spike autosolitions in Gray-Scott model, J. Phys. A-Math. Gen. 48 (2000), 8893-8916. | MR

[19] W.-M. Ni - I. Takagi, Point-condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Industrial Appl. Math. 12 (1995), 327-365. | MR | Zbl

[20] W.-M. Ni - I. Takagi - E. Yanagida, Tohoku Math. J., to appear.

[21] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9-18. | MR | Zbl

[22] Y. Nishiura - D. Ueyama, A skeleton structure of self-replicating dynamics, Physica D. 130 (1999), 73-104. | Zbl

[23] Y. Nishiura, Global structure of bifurcation solutions of some reaction-diffusion systems, SIAM J. Math. Anal. 13 (1982), 555-593. | MR | Zbl

[24] J. Wei, On the construction ofsingle-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996), 315-333. | MR | Zbl

[25] J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems, Int. J. Bifur. and Chaos 10 (2000), 1485-1496. | MR | Zbl

[26] J. Wei - M. Winter, On the two dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal. 30 (1999), 1241-1263. | MR | Zbl

[27] J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness, spectrum estimates and stability analysis, Europ. J. Appl. Math. 10 (1999), 353-378. | MR | Zbl

[28] J. Wei, Existence, stability and metastability of point condensation patterns generated by Gray-Scott system, Nonlinearity 3 (1999), 593-616. | MR | Zbl

[29] J. Wei, Point condensations generated by Gierer-Meinhardt system: a brief survey, book chapter, In: "New Trends in Nonlinear Partial Differential Equations 2000", Y. Morita - H. Ninomiya - E. Yanagida - S. Yotsutani (eds), pp. 46-59.

[30] L. Zhang, Uniqueness of positive solutions of Δu + λu + uP = 0 in a ball, Comm. Partial Differential Equations 17 (1992), 1141-1164. | Zbl