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Journal d’Analyse Mathématique

, Volume 84, Issue 1, pp 1–49 | Cite as

Nonexistence results and estimates for some nonlinear elliptic problems

  • Marie-Francoise Bidaut-Véron
  • Stanislav Pohozaev
Article

Abstract

Here we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type\(L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^\sigma u^Q \) or\(\begin{gathered} L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^\sigma u^S v^R \hfill \\ L_{\mathcal{B}^u } = - div\left[ {\mathcal{B}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^b u^Q u^T \hfill \\ \end{gathered} \). We give integral estimates and nonexistence results. They depend on properties of the supersolutions of the equationsL A u=0,L B v=0, which suppose weak coercivity conditions. Under stronger conditions, we give pointwise estimates in case of equalities, using Harnack properties.

Keywords

Harnack Inequality Nonnegative Solution Radial Solution Integral Estimate Strong Maximum Principle 
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.

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References

  1. [1]
    D. Andreucci, M. A. Herrero and J. J. Velázquez,Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. H. Poincaré Anal. Non Linéaire,14 (1997), 1–52.zbMATHCrossRefGoogle Scholar
  2. [2]
    P. Aviles,Local behaviour of solutions of some elliptic equations, Comm. Math. Phys.108 (1987), 177–192.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    H. Berestycki, I. Capuzzo Dolcetta and L. Niremberg,Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear anal.4 (1993), 59–78.Google Scholar
  4. [4]
    M-F. Bidaut-Véron,Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal.107 (1989), 293–324.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M-F. Bidaut-Véron,Singularities of solutions of a class of quasilinear equations in divergence form, in Nonlinear Diffusion Equations and their Equilibrium States, 3, Birkhaüser, Boston, 1992, pp. 129–144.Google Scholar
  6. [6]
    M-F. Bidaut-Véron,Rotationally symmetric hypersurfaces with prescribed mean curvature, Pacific J. Math.173 (1996), 29–67.zbMATHMathSciNetGoogle Scholar
  7. [7]
    M.-F. Bidaut-Véron,Local behaviour of solutions of a class of nonlinear elliptic systems Adv. Differential Equations 5 (2000), 147–192.zbMATHMathSciNetGoogle Scholar
  8. [8]
    M-F. Bidau-Véron and P. Grillot,Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa28 (1999), 229–271.Google Scholar
  9. [9]
    M-F. Bidaut-Véron and P. Grillot,Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymptotic Anal.19 (1999), 117–147.zbMATHGoogle Scholar
  10. [10]
    M-F. Bidaut-Véron and T. Raoux,Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations21 (1996), 1035–1086.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M-F. Bidaut-Véron and L. Véron,Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math.106 (1991), 489–539.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    M-F. Bidaut-Véron and L. Vivier,An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Rev. Mat. Iberoamericana (to appear).Google Scholar
  13. [13]
    I. Birindelli and E. Mitidieri,Liouville theorems for elliptic inequalities and applications, Proc. Royal Soc. Edinburgh128A (1998), 1217–1247.MathSciNetGoogle Scholar
  14. [14]
    L. Cafarelli, B. Gidas and J. Spruck,Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math.42 (1989), 271–297.CrossRefMathSciNetGoogle Scholar
  15. [15]
    G. Caristi and E. Mitidieri,Nonexistence of solutions of quasilinear equations, Adv. Differential Equations,2 (1997), 319–359.zbMATHMathSciNetGoogle Scholar
  16. [16]
    P. Clément, J. Fleckinger, E. Mitidieri and F. de Thélin,Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations166 (2000), 455–477.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    D. De Figueiredo and P. Felmer,A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa21 (1994), 387–397.zbMATHGoogle Scholar
  18. [18]
    M. Garcia, R. Manasevich, E. Mitidieri, and C. Yarur,Existence and nonexistence of singular positive solutions for a class of semilinear elliptic systems, Arch. Rational Mech. Anal.140 (1997), 253–284.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    B. Gidas and J. SpruckA priori bounds for positive solutions of nonilinear elliptic equations, Comm. Partial Differential Equations6 (1981), 883–901.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    B. Gidas and J. Spruck,Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math.34 (1981), 525–598.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    D. Gilbarg and N. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, New York, 1983.zbMATHGoogle Scholar
  22. [22]
    S. Kichenassamy and L. Véron,Singular solutions of the p-Laplace equation, Math. Ann.275 (1986), 599–615.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    V. Kondratyev and S. Eidelman,Positive solutions of quasilinear Emden-Fowler systems with arbitrary order, Russian J. Math. Phys.2 (1995), 535–540.MathSciNetGoogle Scholar
  24. [24]
    E. Mitidieri and S. Pohozaev,The absence of global positive solutions to quasilinear elliptic inequalities, Doklady Math.57 (2) (1998), 456–460.MathSciNetGoogle Scholar
  25. [25]
    E. Mitidieri and S. Pohozaev,Nonexistence of positive solutions for a system of quasilinear elliptic inequalities Doklady Akad. Nauk366 (1999), 13–17.zbMATHMathSciNetGoogle Scholar
  26. [26]
    E. Mitidieri and S. Pohozaev,Nonexistence of positive solutions for quasilinear elliptic problems inN, Proc. Steklov Inst. Math.227 (to appear).Google Scholar
  27. [27]
    W. Ni and J. Serrin,Nonexistence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo Suppl.5 (1986), 171–185.MathSciNetGoogle Scholar
  28. [28]
    W. Ni and J. Serrin,Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei, Conv. Dei Lincei77 (1986), 231–257.Google Scholar
  29. [29]
    S. Pohozaev,On the eigenfunctions of quasilinear elliptic problems, Math. USSR Sbornik11 (1970), 171–188.CrossRefGoogle Scholar
  30. [30]
    S. Pohozaev,The essentially nonlinear capacities induced by differential operators, Doklady Math.56 (3) (1997), 924–926.Google Scholar
  31. [31]
    P. Pucci and J. Serrin,Continuation and limit properties for solutions of strongly nonlinear second order differential equations, Asymptotic Anal.,4 (1991), 97–160.zbMATHMathSciNetGoogle Scholar
  32. [32]
    J. Serrin,Local behavior of solutions of quasilinear equations, Acta Math.111 (1964), 247–302.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    J. Serrin,Isolated singularities of solutions of quasilinear equations, Acta Math.113 (1965), 219–240.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    J. Serrin and H. Zou,Non-existence of positive solutions for the Lane-Emden system, Differential Integral Equations9 (1996), 635–653.zbMATHMathSciNetGoogle Scholar
  35. [35]
    P. Tolksdorff,Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations51 (1984), 126–150.CrossRefMathSciNetGoogle Scholar
  36. [36]
    N. Trudinger,On Harnack type inequalities and their application to quasilinear equations, Comm. Pure Appl. Math.20 (1967), 721–747.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    J. L. Vazquez,A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim.12 (1984), 191–202.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2001

Authors and Affiliations

  • Marie-Francoise Bidaut-Véron
    • 1
  • Stanislav Pohozaev
    • 2
  1. 1.CNRS UPRES-A 6083 Faculté des SciencesLaboratoire de Mathématiques et Physique ThéoriqueToursFrance
  2. 2.Steklov Mathematical InstituteMoscowRussia

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