Mathematische Annalen

, Volume 333, Issue 2, pp 231–260 | Cite as

Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems

  • Djairo G. de Figueiredo
  • Boyan Sirakov


Elliptic System Type Theorem Liouville Type Liouville Type Theorem Monotonicity Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexandrov, A.D.: A characteristic property of the spheres. Ann. Mat. Pura Appl. 58, 303–354, (1962)Google Scholar
  2. 2.
    Alves, C., de Figueiredo, D.G.: Nonvariational elliptic systems. Discr. Cont. Dyn. Systems 8(2), 289–302 (2002)Google Scholar
  3. 3.
    Amann, H.: Fixed point equation and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 18, 620–709 (1976)CrossRefGoogle Scholar
  4. 4.
    Amann, H.: On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal. 14, 349–381 (1973)CrossRefGoogle Scholar
  5. 5.
    Bandle, C., Essen, M.: On positive solutions of Emden equations in cones. Arch. Rat. Mech. Anal. 112(4), 319–338 (1990)Google Scholar
  6. 6.
    Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 69–94 (1997)Google Scholar
  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(1), 47–92 (1994)Google Scholar
  8. 8.
    Birindelli, I., Mitidieri, E.: Liouville theorems for elliptic inequalities and inequations. Proc. Royal Soc. Edinburgh, 128A, 1217–1247 (1998)Google Scholar
  9. 9.
    Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bull. Soc. Brazil Mat. Nova Ser 22, 1–37 (1991)Google Scholar
  10. 10.
    Busca, J., Manasevich, R.: A Liouville type theorem for Lane-Emden systems. Indiana Univ. Math. J 51(1), 37–51 (2002)Google Scholar
  11. 11.
    Busca, J., Sirakov, B.: Symmetry results for semilinear elliptic systems in the whole space. J. Diff. Eq 163(1), 41–56 (2000)CrossRefGoogle Scholar
  12. 12.
    Busca, J., Sirakov, B.: Harnack type estimates for nonlinear elliptic systems and applications. Ann. Inst. H. Poincare, Anal. Non. Linéaire 21, 543–590 (2005)Google Scholar
  13. 13.
    Cabre, X.: On the Alexandrov-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Comm. Pure and Appl. Math 48, 539–570 (1995)Google Scholar
  14. 14.
    Cabre, X.: Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discr. Cont. Dyn. Syst 8(2), 331–359 (2002)Google Scholar
  15. 15.
    Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations, Duke Math. J 63, 615–623 (1991)MathSciNetGoogle Scholar
  16. 16.
    Clement, Ph., de Figueiredo, D.G., Mitidieri, E.: Positive solutions of semilinear elliptic systems. Comm. Part. Diff. Eq 17, 923–940 (1992)Google Scholar
  17. 17.
    Dancer, E.N.: Some notes on the method of moving planes. Bull. Austral. Math. Soc 46, 425–434 (1992)Google Scholar
  18. 18.
    de Figueiredo, D.G.: Monotonicity and symmetry of solutions of elliptic systems in general domains. NoDEA 1, 119–123 (1994)CrossRefGoogle Scholar
  19. 19.
    de Figueiredo, D.G.: Semilinear elliptic systems. Nonl. Funct. Anal. Appl. Diff. Eq. World Sci. Publishing, River Edge, 1998 pp. 122–152Google Scholar
  20. 20.
    de Figueiredo, D.G., Felmer, P.: A Liouville-type theorem for elliptic systems. Ann. Sc. Norm. Sup. Pisa 21, 387–397 (1994)Google Scholar
  21. 21.
    de Figueiredo, D.G., Lions, P.-L., Nussbaum, R.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pures Appl 61, 41–63 (1982)Google Scholar
  22. 22.
    Gidas, B.: Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations. Lect. Notes on Pure Appl. Math 54, 255–273 (1980)Google Scholar
  23. 23.
    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys 6, 883–901 (1981)Google Scholar
  24. 24.
    Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Part. Diff. Eq. (6), 883–901 (1981)Google Scholar
  25. 25.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edition, Revised Third Printing, Springer Berlin Heidelberg 1998Google Scholar
  26. 26.
    Hulshof, J., van der Vorst, R.C.A.M.: Differential systems with strongly indefinite variational structure. J. Funct. Anal 114, 32–58 (1993)CrossRefGoogle Scholar
  27. 27.
    Krasnoselskii, M.A.: Positive solutions of operator equations, P. Noordhoff, Groningen.Google Scholar
  28. 28.
    Krylov: Nonlinear elliptic and parabolic equations of second order. Coll. Math. and its Appl. (1987)Google Scholar
  29. 29.
    Laptev, G.G.: Absence of global positive solutions of systems of semilinear elliptic inequalities in cones. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 64(6), 107–124, (2000); translation in Izv. Math. 64(6), 1197–1215 (2000)Google Scholar
  30. 30.
    Mitidieri, E.: Non-existence of positive solutions of semilinear elliptic systems in ℝN. Quaderno Matematico 285, (1992)Google Scholar
  31. 31.
    Mitidieri, E.: Non-existence of positive solutions of semilinear elliptic systems in ℝN. Diff. Int. Eq 9(3), 465–479 (1996)Google Scholar
  32. 32.
    Mitidieri, E., Pohozaev, S.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 1–384 (2001)Google Scholar
  33. 33.
    Montenegro, M.S.: Criticalidade, superlinearidade e sublinearidade para sistemas elípticos semilineares, Tese de Doutoramento, Unicamp (1997)Google Scholar
  34. 34.
    Nussbaum, R.: Positive solutions of nonlinear elliptic boundary value problems. J. Math. Anal. Appl 51, 461–482 (1975)CrossRefGoogle Scholar
  35. 35.
    Serrin, J.: A symmetry theorem in potential theory. Arch. Rat. Mech. Anal 43, 304–318 (1971)CrossRefGoogle Scholar
  36. 36.
    Serrin, J., Zou, H.: Non-existence of positive solutions of Lane-Emden systems. Diff. Int. Eq. 9(4) (1996), 635–653.Google Scholar
  37. 37.
    Sirakov, B.: Notions of sublinearity and superlinearity for nonvariational elliptic systems. Discrete Cont. Dyn. Syst. A13, 163–174 (2005)Google Scholar
  38. 38.
    Souto, M.A.S.: A priori estimates and and existence of positive solutions of nonlinear cooperative elliptic systems. Diff. Int. Eq. 8(5), 1245–1258 (1995)Google Scholar
  39. 39.
    Sweers, G.: Strong positivity in Open image in new window for elliptic systems. Math. Z. 209, 251–271 (1992)Google Scholar
  40. 40.
    Zou, H.: A priori estimates for a semilinear elliptic system without variational structure and their application. Math. Ann 323, 713–735 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Djairo G. de Figueiredo
    • 1
  • Boyan Sirakov
    • 2
    • 3
  1. 1.IMECC-UNICAMPCampinasBrazil
  2. 2.Laboratoire MODALX, UFR SEGMINanterre CedexFrance
  3. 3.CAMS, EHESSParis Cedex 06France

Personalised recommendations