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On positive solutions of general nonlinear elliptic symbiotic interacting systems. (English) Zbl 0757.35023
Summary: After explaining the background of the general symbiotic interactions, we give the sufficient and necessary conditions for the existence of positive solutions to the general elliptic symbiotic models of Dirichlet problem. The result is that such existence of positive solution is characterized by the positive constant equilibria of this system. We also prove that there could be any number of solutions to the symbiotic systems as a result of a singular perturbation of the generators of \(C_ 0\)-semigroup in Banach spaces.

MSC:
35J60 Nonlinear elliptic equations
35P05 General topics in linear spectral theory for PDEs
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[1] DOI: 10.1137/1018114 · Zbl 0345.47044 · doi:10.1137/1018114
[2] Blat J., Proc. Roy. Soc. Edinburgh, (A) 97 pp 21– (1984)
[3] Cantree U.S., to appear, Huston J. of Math 97 (1984)
[4] Dancer E.N., On the existence and uniqueness of positive solutions for competing species models with diffusions (1989)
[5] Goldstein J.A., Semigroups of Linear Operators and Applications (1985) · Zbl 0592.47034
[6] Kato T., Perturbation Theory for Linear Operators (1984) · Zbl 0531.47014
[7] Korman P., Proc. Roy. Soc. Edinburgh 102 pp 315– (1986)
[8] DOI: 10.1080/00036818708839706 · Zbl 0639.35026 · doi:10.1080/00036818708839706
[9] DOI: 10.1090/S0002-9947-1988-0920151-1 · doi:10.1090/S0002-9947-1988-0920151-1
[10] Li L., Proc. Roy. Soc. Edinburgh 110 pp 295– (1988)
[11] Li L., Proc. Amer. Math. Soc. (to appear) 110 (1988)
[12] DOI: 10.1080/00036818608839592 · Zbl 0593.35042 · doi:10.1080/00036818608839592
[13] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[14] Roitt I., Immunology (1986)
[15] Vainberg M.M., Variational Methods for the Study of Nonlinear Operators (1984)
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