The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems

Journal of Differential Equations

Volumn 127, Issue 1, 1996, Pages 263-294

  López Gómez, Julían ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030140612     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1996.0070     Document Type: Article

References (25)

1. 1. H. AMANN, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709.
2. 2. J. M. ARRIETA, J. K. HALE, AND Q. HAN, Eigenvalue problems for nonsmoothly perturbed domains, J. Differential Equations 91 (1991), 24-52.
3. 3. I. BABUSKA AND R. VYBORNY, Continuous dependence of eigenvalues on the domain, Czech. Math. J. 15 (1965), 169-178.
4. 4. H. BERESTYCKI AND L. NIRENBERG, On the method of moving planes and the sliding method, Boll. Soc. Bras. Mat. Nov. Ser. 22 (1991), 1-37.
5. 5. H. BERESTYCKI, L. NIRENBERG, AND S. R. S. VARADHAN, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math. XLVII (1994), 47-92.
6. 6. K. J. BROWN AND C. C. LIN, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight-function, J. Math. Anal. Appl. 75 (1980), 112-120.
7. 7. R. COURANT AND D. HILBERT, "Methods of Mathematical Physics," Vol. I, II, Wiley-Interscience, New York, 1962.
8. 8. M. G. CRANDALL AND P. H. RABINOWITZ, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational. Mech. Anal. 67 (1973), 161-180.
9. 9. E. N. DANCER, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984), 729-743.
10. 10. E. N. DANCER, The effect of domain shape on the number of positive solutions of certain nonlinear equations, II, J. Differential Equations 87 (1990), 316-339.
11. 11. C. FABER, Beweis das unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math. Phys. (1923), 169-172.
12. 12. D. GILBARG AND N. S. TRUDINGER, "Elliptic Partial Differential Equations of the Second Order," Springer, Berlin 1977.
13. 13. J. P. GOSSEZ AND E. LAMI-DOZO, On the principal eigenvalue of a second order linear elliptic problem, Arch. Rational Mech. Anal. 89 (1985), 169-175.
14. 14. P. HESS, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics, Vol. 247, Longman Sci. Tech., Harlow, 1991.
15. 15. P. HESS AND T. KATO, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), 999-1030.
16. 16. T. KATO, Superconvexity of the spectral radius and convexity of the spectral bound and type, Math. Z. 180 (1982), 265-273.
17. 17. E. KRAHN, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 91 (1925), 97-100.
18. 18. E. H. LIEB, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), 441-448.
19. 19. J. LÓPEZ-GÓMEZ, On linear weighted boundary value problems, in "Partial Differential Equations; Models in Physics and Biology" (G. Lumer, S. Nicaise, and B. W. Schulze, Eds.), pp. 188-203, Akademie-Verlag, Berlin, 1994.
20. 20. J. LÓPEZ-GÓMEZ AND J. C. SABINA, Coexistence states and global attractivity for some convective diffusive competing species models, Trans. Amer. Math. Soc. 347 (1995), 3797-3833.
21. 21. A. MANES AND A. M. MICHELETTI, Un'estensione della teoria variazonale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. 7 (1973), 285-301.
22. 22. M. H. PROTTER AND H. F. WEINBERGER, On the spectrum of general second order operators, Bull. Amer. Math. Soc. 72 (1966), 251-255.
23. 23. M. H. PROTTER AND H. F. WEINBERGER, "Maximum Principles in Differential Equations," Prentice-Hall, Englewood Cliffs, NJ, 1967.
24. 24. W. WALTER, A theorem on elliptic inequalities with an application to gradient bounds, Math. Z. 200 (1989), 293-299.
25. 25. J. WLOKA, "Partial Differential Equations," Cambridge Univ. Press, Cambridge, UK, 1987.