@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}, volume = {19}, year = {2002}, pages = {777-813}, mrnumber = {1939086}, zbl = {1020.35065}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2002__19_6_777_0} }
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) pp. 777-813. http://gdmltest.u-ga.fr/item/AIHPC_2002__19_6_777_0/
[1] Sobolev Spaces, Academic Press, 1975. | MR 450957 | Zbl 0314.46030
,[2] 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 125307 | Zbl 0093.10401
, , ,[3] Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976) 620-709. | MR 415432 | Zbl 0345.47044
,[4] Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519-543. | MR 1276168 | Zbl 0805.35028
, , ,[5] Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Analysis TMA 16 (1982) 1001-1022. | MR 678053 | Zbl 0518.35050
, , ,[6] The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 303 (1987) 487-501. | MR 902780 | Zbl 0633.35041
, , ,[7] The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47-92. | MR 1258192 | Zbl 0806.35129
, , ,[8] The principle of linearized stability for a class of degenerate diffusion equations, J. Differential Equations 57 (1985) 373-405. | MR 790282 | Zbl 0583.35061
, ,[9] Sublinear elliptic equations in L1, Houston Math. J. 20 (1994) 99-114. | MR 1272564 | Zbl 0803.35047
, ,[10] Sublinear elliptic equations in Rn, Manuscripta Math. 74 (1992) 87-106. | MR 1141779 | Zbl 0761.35027
, ,[11] Remarks on sublinear elliptic equations, Nonlinear Analysis TMA 10 (1986) 55-64. | MR 820658 | Zbl 0593.35045
, ,[12] Methods of Bifurcation Theory, Springer-Verlag, 1982. | MR 660633 | Zbl 0487.47039
, ,[13] A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities, Pitman Research Notes 343 (1996) 73-91. | MR 1417272 | Zbl 0868.35012
, , ,[14] Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970) 217-226. | MR 259356 | Zbl 0201.43102
, ,[15] An introduction to constructive aspects of bifurcation and the implicit function theorem, in: (Ed.), Applications of Bifurcation Theory, Academic Press, New York, 1977, pp. 1-35. | MR 463988
,[16] Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321-340. | MR 288640 | Zbl 0219.46015
, ,[17] Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal. 52 (1973) 161-180. | MR 341212 | Zbl 0275.47044
, ,[18] On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977) 193-222. | MR 427826 | Zbl 0362.35031
, , ,[19] Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Springer-Verlag, Berlin, 1990. | MR 1064315 | Zbl 0766.47001
, ,[20] Nonlinear Partial Differential Equations and Free Boundaries, Pitman, Boston, 1985. | MR 908901 | Zbl 0595.35100
,[21] Foundations of Modern Analysis, Academic Press, New York, 1960. | MR 120319 | Zbl 0100.04201
,[22] On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math. 29 (1976) 595-621. | MR 425380 | Zbl 0356.35065
, ,[23] 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 49.0342.03
,[24] A singular nonlinear equation, Osaka J. Math. 12 (1960) 1-19. | MR 123095 | Zbl 0097.30202
, ,[25] The regulation of inhomogeneous populations, J. Theor. Biol. 52 (1975) 441-457.
, ,[26] On the diffusion of biological populations, Math. Biosci. 33 (1977) 35-49. | MR 682594 | Zbl 0362.92007
, ,[27] Geometric Theory of Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin, 1981. | MR 610244 | Zbl 0456.35001
,[28] J. Hernández, in preparation, 1999.
[29] On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980) 999-1030. | MR 588690 | Zbl 0477.35075
, ,[30] Development of Aleksandrov's theory of the isotropic strict extremum principle, Differential Equations (English translation) 16 (1980) 181-189. | Zbl 0448.35020
, ,[31] Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. Ann. 91 (1925) 97-100. | JFM 51.0356.05 | MR 1512244
,[32] Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968. | MR 241822 | Zbl 0174.15403
, , ,[33] Uniqueness of sublinear boundary value problems, J. Differential Equations 13 (1973) 13-23. | MR 369899 | Zbl 0247.35052
,[34] The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996) 263-294. | MR 1387266 | Zbl 0853.35078
,[35] Un estensione della teoria variazonale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. 7 (1973) 285-301. | MR 344663 | Zbl 0275.49042
, ,[36] Density-dependent dispersal and spatial distribution of a population, J. Theor. Biol. 86 (1980) 351-363. | MR 585955
,[37] Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. | MR 1212084 | Zbl 0777.35001
,[38] Maximum Principles in Differential Equations, Springer-Verlag, Berlin, 1984. | MR 762825 | Zbl 0549.35002
, ,[39] 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 0088.30501
,[40] Théorie du degré topologique et applications à des problèmes aux limites non linéaires, Lecture Notes, Lab. Analyse Numérique, Univ. Paris VI, Paris, 1975.
,[41] Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972) 979-1000. | MR 299921 | Zbl 0223.35038
,[42] Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984) 1-29. | MR 726104 | Zbl 0554.35064
,[43] Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983. | MR 688146 | Zbl 0508.35002
,[44] Uniqueness of a diffusion model of population biology, Comm. Partial Differential Equations 8 (1983) 1605-1620. | MR 729195 | Zbl 0534.35055
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