On the linearization of some singular, nonlinear elliptic problems and applications
Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 6, pp. 777-813.
@article{AIHPC_2002__19_6_777_0,
     author = {Hern\'andez, Jes\'us and Mancebo, Francisco J and Vega, Jos\'e M},
     title = {On the linearization of some singular, nonlinear elliptic problems and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {777--813},
     publisher = {Elsevier},
     volume = {19},
     number = {6},
     year = {2002},
     mrnumber = {1939086},
     zbl = {1020.35065},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2002__19_6_777_0/}
}
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Hernández, Jesús; Mancebo, Francisco J; Vega, José M. On the linearization of some singular, nonlinear elliptic problems and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 6, pp. 777-813. http://www.numdam.org/item/AIHPC_2002__19_6_777_0/

[1] Adams R.A., Sobolev Spaces, Academic Press, 1975. | MR | Zbl

[2] Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959) 623-727. | MR | Zbl

[3] Amann H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976) 620-709. | MR | Zbl

[4] Ambrosetti A., Brezis H., Cerami G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519-543. | MR | Zbl

[5] Aronson D., Crandall M.G., Peletier L.A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Analysis TMA 16 (1982) 1001-1022. | MR | Zbl

[6] Bandle C., Pozio M.A., Tesei A., The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 303 (1987) 487-501. | MR | Zbl

[7] Berestycki H., Nirenberg L., Varadhan S.R.S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47-92. | MR | Zbl

[8] Bertsch M., Rostamian R., The principle of linearized stability for a class of degenerate diffusion equations, J. Differential Equations 57 (1985) 373-405. | MR | Zbl

[9] Boccardo L., Orsina L., Sublinear elliptic equations in L1, Houston Math. J. 20 (1994) 99-114. | MR | Zbl

[10] Brezis H., Kamin S., Sublinear elliptic equations in Rn, Manuscripta Math. 74 (1992) 87-106. | MR | Zbl

[11] Brezis H., Oswald L., Remarks on sublinear elliptic equations, Nonlinear Analysis TMA 10 (1986) 55-64. | MR | Zbl

[12] Chow S.N., Hale J.K., Methods of Bifurcation Theory, Springer-Verlag, 1982. | MR | Zbl

[13] Clément Ph., De Figueiredo D.G., Mitidieri E., A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities, Pitman Research Notes 343 (1996) 73-91. | MR | Zbl

[14] Cohen D., Laetsch T., Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970) 217-226. | MR | Zbl

[15] Crandall M.G., An introduction to constructive aspects of bifurcation and the implicit function theorem, in: Rabinowitz P.H. (Ed.), Applications of Bifurcation Theory, Academic Press, New York, 1977, pp. 1-35. | MR

[16] Crandall M.G., Rabinowitz P.H., Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321-340. | MR | Zbl

[17] Crandall M.G., Rabinowitz P.H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal. 52 (1973) 161-180. | MR | Zbl

[18] Crandall M.G., Rabinowitz P.H., Tartar L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977) 193-222. | MR | Zbl

[19] Dautray R., Lions J.L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Springer-Verlag, Berlin, 1990. | MR | Zbl

[20] Díaz J.I., Nonlinear Partial Differential Equations and Free Boundaries, Pitman, Boston, 1985. | MR | Zbl

[21] Dieudonné J., Foundations of Modern Analysis, Academic Press, New York, 1960. | MR | Zbl

[22] Donsker M., Varadhan S.R.S., On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math. 29 (1976) 595-621. | MR | Zbl

[23] Faber C., Beweis das unter allen homogenen membranen von gleicher fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt, Sitzunsber, Bayer. Akad. der Wiss. Math. Phys. (1923) 169-172. | JFM

[24] Fulks W., Maybee J.S., A singular nonlinear equation, Osaka J. Math. 12 (1960) 1-19. | MR | Zbl

[25] Gurney W.S.C., Nisbet R.N., The regulation of inhomogeneous populations, J. Theor. Biol. 52 (1975) 441-457.

[26] Gurtin M.E., Maccamy R.C., On the diffusion of biological populations, Math. Biosci. 33 (1977) 35-49. | MR | Zbl

[27] Henry D., Geometric Theory of Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin, 1981. | MR | Zbl

[28] J. Hernández, in preparation, 1999.

[29] Hess P., Kato T., On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980) 999-1030. | MR | Zbl

[30] Kamynin L.I., Khimchenko B.N., Development of Aleksandrov's theory of the isotropic strict extremum principle, Differential Equations (English translation) 16 (1980) 181-189. | Zbl

[31] Krahn E., Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. Ann. 91 (1925) 97-100. | JFM | MR

[32] Ladyženskaja O.A., Solonnikov V.A., Ural'Ceva N.N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968. | MR | Zbl

[33] Laetsch T., Uniqueness of sublinear boundary value problems, J. Differential Equations 13 (1973) 13-23. | MR | Zbl

[34] López-Gómez J., The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996) 263-294. | MR | Zbl

[35] Manes A., Micheletti A.M., Un estensione della teoria variazonale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. 7 (1973) 285-301. | MR | Zbl

[36] Namba T., Density-dependent dispersal and spatial distribution of a population, J. Theor. Biol. 86 (1980) 351-363. | MR

[37] Pao C.V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. | MR | Zbl

[38] Protter M.H., Weinberger H.F., Maximum Principles in Differential Equations, Springer-Verlag, Berlin, 1984. | MR | Zbl

[39] Pucci C., Propietà di massimo e minimo delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico, Atti. Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. 23 (1957) 370-375. | Zbl

[40] Rabinowitz P.H., Théorie du degré topologique et applications à des problèmes aux limites non linéaires, Lecture Notes, Lab. Analyse Numérique, Univ. Paris VI, Paris, 1975.

[41] Sattinger D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972) 979-1000. | MR | Zbl

[42] Schatzman M., Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984) 1-29. | MR | Zbl

[43] Smoller J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983. | MR | Zbl

[44] Spruck J., Uniqueness of a diffusion model of population biology, Comm. Partial Differential Equations 8 (1983) 1605-1620. | MR | Zbl