On the linearization of some singular, nonlinear elliptic problems and applications

Jesús Hernández; Francisco J Mancebo; José M Vega

Annales de l'I.H.P. Analyse non linéaire (2002)

  • Volume: 19, Issue: 6, page 777-813
  • ISSN: 0294-1449

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Hernández, Jesús, Mancebo, Francisco J, and Vega, José M. "On the linearization of some singular, nonlinear elliptic problems and applications." Annales de l'I.H.P. Analyse non linéaire 19.6 (2002): 777-813. <http://eudml.org/doc/78562>.

@article{Hernández2002,
author = {Hernández, Jesús, Mancebo, Francisco J, Vega, José M},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {semilinear problem; linearization; spectrum; existence; uniqueness; principal eigenvalue; Green operator; Fréchet differentiability},
language = {eng},
number = {6},
pages = {777-813},
publisher = {Elsevier},
title = {On the linearization of some singular, nonlinear elliptic problems and applications},
url = {http://eudml.org/doc/78562},
volume = {19},
year = {2002},
}

TY - JOUR
AU - Hernández, Jesús
AU - Mancebo, Francisco J
AU - Vega, José M
TI - On the linearization of some singular, nonlinear elliptic problems and applications
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2002
PB - Elsevier
VL - 19
IS - 6
SP - 777
EP - 813
LA - eng
KW - semilinear problem; linearization; spectrum; existence; uniqueness; principal eigenvalue; Green operator; Fréchet differentiability
UR - http://eudml.org/doc/78562
ER -

References

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  1. [1] Adams R.A., Sobolev Spaces, Academic Press, 1975. Zbl0314.46030MR450957
  2. [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. Zbl0093.10401MR125307
  3. [3] Amann H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev.18 (1976) 620-709. Zbl0345.47044MR415432
  4. [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. Zbl0805.35028MR1276168
  5. [5] Aronson D., Crandall M.G., Peletier L.A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Analysis TMA16 (1982) 1001-1022. Zbl0518.35050MR678053
  6. [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. Zbl0633.35041MR902780
  7. [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. Zbl0806.35129MR1258192
  8. [8] Bertsch M., Rostamian R., The principle of linearized stability for a class of degenerate diffusion equations, J. Differential Equations57 (1985) 373-405. Zbl0583.35061MR790282
  9. [9] Boccardo L., Orsina L., Sublinear elliptic equations in L1, Houston Math. J.20 (1994) 99-114. Zbl0803.35047MR1272564
  10. [10] Brezis H., Kamin S., Sublinear elliptic equations in Rn, Manuscripta Math.74 (1992) 87-106. Zbl0761.35027MR1141779
  11. [11] Brezis H., Oswald L., Remarks on sublinear elliptic equations, Nonlinear Analysis TMA10 (1986) 55-64. Zbl0593.35045MR820658
  12. [12] Chow S.N., Hale J.K., Methods of Bifurcation Theory, Springer-Verlag, 1982. Zbl0487.47039MR660633
  13. [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 Notes343 (1996) 73-91. Zbl0868.35012MR1417272
  14. [14] Cohen D., Laetsch T., Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations7 (1970) 217-226. Zbl0201.43102MR259356
  15. [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. MR463988
  16. [16] Crandall M.G., Rabinowitz P.H., Bifurcation from simple eigenvalues, J. Funct. Anal.8 (1971) 321-340. Zbl0219.46015MR288640
  17. [17] Crandall M.G., Rabinowitz P.H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal.52 (1973) 161-180. Zbl0275.47044MR341212
  18. [18] Crandall M.G., Rabinowitz P.H., Tartar L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations2 (1977) 193-222. Zbl0362.35031MR427826
  19. [19] Dautray R., Lions J.L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Springer-Verlag, Berlin, 1990. Zbl0766.47001MR1064315
  20. [20] Díaz J.I., Nonlinear Partial Differential Equations and Free Boundaries, Pitman, Boston, 1985. Zbl0595.35100MR908901
  21. [21] Dieudonné J., Foundations of Modern Analysis, Academic Press, New York, 1960. Zbl0100.04201MR120319
  22. [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. Zbl0356.35065MR425380
  23. [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. Zbl49.0342.03JFM49.0342.03
  24. [24] Fulks W., Maybee J.S., A singular nonlinear equation, Osaka J. Math.12 (1960) 1-19. Zbl0097.30202MR123095
  25. [25] Gurney W.S.C., Nisbet R.N., The regulation of inhomogeneous populations, J. Theor. Biol.52 (1975) 441-457. 
  26. [26] Gurtin M.E., MacCamy R.C., On the diffusion of biological populations, Math. Biosci.33 (1977) 35-49. Zbl0362.92007MR682594
  27. [27] Henry D., Geometric Theory of Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin, 1981. Zbl0456.35001MR610244
  28. [28] J. Hernández, in preparation, 1999. 
  29. [29] Hess P., Kato T., On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations5 (1980) 999-1030. Zbl0477.35075MR588690
  30. [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. Zbl0448.35020
  31. [31] Krahn E., Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. Ann.91 (1925) 97-100. Zbl51.0356.05MR1512244JFM51.0356.05
  32. [32] Ladyženskaja O.A., Solonnikov V.A., Ural'ceva N.N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968. Zbl0174.15403MR241822
  33. [33] Laetsch T., Uniqueness of sublinear boundary value problems, J. Differential Equations13 (1973) 13-23. Zbl0247.35052MR369899
  34. [34] López-Gómez J., The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations127 (1996) 263-294. Zbl0853.35078MR1387266
  35. [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. Zbl0275.49042MR344663
  36. [36] Namba T., Density-dependent dispersal and spatial distribution of a population, J. Theor. Biol.86 (1980) 351-363. MR585955
  37. [37] Pao C.V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Zbl0777.35001MR1212084
  38. [38] Protter M.H., Weinberger H.F., Maximum Principles in Differential Equations, Springer-Verlag, Berlin, 1984. Zbl0549.35002MR762825
  39. [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. Zbl0088.30501
  40. [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. [41] Sattinger D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J.21 (1972) 979-1000. Zbl0223.35038MR299921
  42. [42] Schatzman M., Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J.33 (1984) 1-29. Zbl0554.35064MR726104
  43. [43] Smoller J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983. Zbl0508.35002MR688146
  44. [44] Spruck J., Uniqueness of a diffusion model of population biology, Comm. Partial Differential Equations8 (1983) 1605-1620. Zbl0534.35055MR729195

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