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A singular Gierer-Meinhardt system of elliptic equations: the classical case. (English) Zbl 1140.35408
Summary: This paper studies the existence of solutions of a nonlinear elliptic system usually called the generalized Gierer-Meinhardt equations \[ \begin{cases} d_ {1}\Delta u-\alpha_ {1}u+\gamma_ {1}\frac{u^ {r}}{v}=0,\\ d_ {2}\Delta v-\alpha_ {2}v+\gamma_ {2}u^ {r}=0,\\ u| _ {\partial\Omega}=0,v| _ {\partial\Omega}=0.\\ \end{cases} \] This type of system originally arose in studies of pattern-formation in biology, and has interesting and challenging mathematical properties, especially with Dirichlet boundary conditions when the nonlinear terms become singular near the boundary. We study the existence of solutions in the classical case of Gierer-Meinhardt [A. Gierer and H. Meinhardt, Kybernetik 12, No. 1, 30–39 (1972)], when the activator-inhibitor model has different sources.

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations
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[1] M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1972.
[2] Bowman, F., Introduction to Bessel functions, (1958), Dover New York · JFM 64.1087.01
[3] H. Brezis, X. Cabre, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 1 (8,2) (1998) 223-262, or download that from http://www.ann.jussieu.fr/publications/1997/R97026.html · Zbl 0907.35048
[4] C̆anić, S.; Lee Keyfitz, B., An elliptic problem arising form the unsteady transonic small disturbance equation, J. differential equations, 125, 548-574, (1996) · Zbl 0869.35043
[5] Choi, Y.S.; Kim, E.H., On the existence of positive solutions of quasilinear elliptic boundary value problems, J. differential equations, 155, 423-442, (1999) · Zbl 0946.35033
[6] Choi, Y.S.; Lazer, A.C.; McKenna, P.J., On a singular quasilinear anisotropic elliptic boundary value problem, Trans. AMS, 347, 2633-2641, (1995) · Zbl 0835.35049
[7] Choi, Y.S.; McKenna, P.J., On a singular quasilinear anisotropic elliptic boundary value problem, II, Trans. AMS, 350, 2925-2937, (1998) · Zbl 0901.35031
[8] Choi, Y.S.; McKenna, P.J., A singular gierer – meinhardt system of elliptic equations, Ann. inst. Henri poincare, analyse non lineaire, 17, 4, 503-522, (2000) · Zbl 0969.35062
[9] Crandall, M.G.; Rabinowitz, P.H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. PDE, 2, 2, 193-222, (1977) · Zbl 0362.35031
[10] del Pino, M.A., Radially symmetric internal layers in a semilinear elliptic system, Trans. AMS, 347, 4807-4837, (1995) · Zbl 0853.35009
[11] Doelman, A.; Gardner, R.; Kaper, T., Large stable pulse solutions in reaction – diffusion equations, Indiana univ. math. J., 50, 1, 443-507, (2001) · Zbl 0994.35058
[12] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Kyberntik, 12, 30-39, (1972)
[13] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0691.35001
[14] Gui, C., Multipeak solutions for a semilinear Neumann problem, Duke math. J., 84, 739-769, (1996) · Zbl 0866.35039
[15] Lair, A.; Wood, A., Existence of entire solutions of semilinear elliptic systems, J. differential equations, 164, 380-394, (2000) · Zbl 0962.35052
[16] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. AMS, 111, 721-730, (1991) · Zbl 0727.35057
[17] Ni, W.-M.; Polačik, P.; Yanagida, E., Monotonicity of stable solutions in shadow systems, Trans. AMS, 353, 12, 5057-5069, (2001) · Zbl 0981.35018
[18] Ni, W.-M.; Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. pure appl. math., 44, 819-851, (1991) · Zbl 0754.35042
[19] Ni, W.-M.; Takagi, I., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke math. J., 70, 247-281, (1993) · Zbl 0796.35056
[20] Ni, W.-M.; Takagi, I.; Wei, J., On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problemintermediate solutions, Duke math. J., 94, 597-618, (1998) · Zbl 0946.35007
[21] Serrin, J.; Zou, H., Existence of positive entire solutions of elliptic Hamiltonian systems, Comm. partial differential equations, 23, 577-599, (1998) · Zbl 0906.35033
[22] Stuart, C.A., Existence theorems for a class of nonlinear integral equations, Math. Z., 137, 49-66, (1974) · Zbl 0289.45013
[23] Taliaferro, S.D., A nonlinear singular boundary value problem, Nonlinear analysis TMA, 3, 897-904, (1979) · Zbl 0421.34021
[24] P. Turchin, J.D. Reeve, J.T. Cronin, R.T. Wilkens, Spatial pattern formation in ecological systems: bridging theoretical and empirical approaches, in: J. Bascompte, R.V. Sole (Eds.), Modelling Spatiotemporal Dynamics in Ecology, Landes Bioscience, Austin, TX, 1997, pp. 195-210.
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