Archive for Rational Mechanics and Analysis

, Volume 188, Issue 1, pp 155–179 | Cite as

Non-existence of Entire Solutions of Degenerate Elliptic Inequalities with Weights


Non-existence results for non-negative distribution entire solutions of singular quasilinear elliptic differential inequalities with weights are established. Such inequalities include the capillarity equation with varying gravitational field h, as well as the general p-Poisson equation of radiative cooling with varying heat conduction coefficient g and varying radiation coefficient h. Since we deal with inequalities and positive weights, it is not restrictive to assume h radially symmetric. Theorem 1 extends in several directions previous results and says that solely entire large solutions can exist, while Theorem 2 shows that in the p-Laplacian case positive entire solutions cannot exist. The results are based on some qualitative properties of independent interest.


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  1. 1.
    Batt J., Faltenbacher W., Horst E. (1986) Stationary spherically symmetric models in stellar dynamics. Arch. Ration. Mech. Anal. 93, 159–183MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Calzolari E., Filippucci R., Pucci P. (2006) Existence of radial solutions for the p-Laplacian elliptic equations with weights. Discret. Cont. Dyn. Syst. 15, 447–479MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Conley C.H., Pucci P., Serrin J. (2005) Elliptic equations and products of positive definite matrices. Math. Nachrichten 278, 1490–1508MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Lair A.V. (1999) A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations. J. Math. Anal. Appl. 240, 205–218MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lair A.V., Wood A.W. (1999) Large solutions of semilinear elliptic problems. Nonlin. Anal. 37, 805–812CrossRefMathSciNetGoogle Scholar
  6. 6.
    Matukuma T. (1938) The cosmos. Iwanami Shoten, TokyoGoogle Scholar
  7. 7.
    Naito Y., Usami H. (1997) Nonexistence results of positive entire solutions for quasilinear elliptic inequalities. Can. Math. Bull. 40, 244–253MATHMathSciNetGoogle Scholar
  8. 8.
    Pigola S., Rigoli M., Setti A.G. (2003) Volume growth, “a priori” estimates, and geometric applications. Geom. Funct. Anal. 13, 1302–1328MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pigola S., Rigoli M., Setti A.G. (2005) Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 822, 99MathSciNetGoogle Scholar
  10. 10.
    Pucci P., Garcìa-Huidobro M., Manàsevich R., Serrin J. (2006) Qualitative properties of ground states for singular elliptic equations with weights. Annali Mat. Pura Appl. 185, 205–243CrossRefGoogle Scholar
  11. 11.
    Pucci P., Rigoli M. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete Contin. Dyn. Syst. (DCDS-A), p. 21 (2007, to appear)Google Scholar
  12. 12.
    Pucci, P., Sciunzi, B., Serrin, J.: Partial and full symmetry of solutions of quasilinear elliptic equations, via the Comparison Principle. Contemp. Math., special volume dedicated to H. Brezis, 9 p. (2007, to appear)Google Scholar
  13. 13.
    Pucci P., Serrin J. (1998) Uniqueness of ground states for quasilinear elliptic equations in the exponential case. Indiana Univ. Math. J. 47, 529–539MATHMathSciNetGoogle Scholar
  14. 14.
    Pucci P., Serrin J.: The strong maximum principle revisited. J. Diff. Equ. 196, 1–66 (2004), Erratum, J. Differ. Equ. 207, 226–227 (2004)Google Scholar
  15. 15.
    Pucci P., Serrin J. (2006) Dead cores and bursts for quasilinear singular elliptic equations. SIAM J. Math. Anal. 38, 259–278MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pucci P., Serrin J. (2007) Maximum principles for elliptic partial differential equations. In: Chipot M.(ed) Handbook of Differential Equations—Stationary Partial Differential Equations, vol 4. Elsevier BV, Amsterdam, pp. 355–483CrossRefGoogle Scholar
  17. 17.
    Pucci, P., Servadei, R.: Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights. Ann. Inst. H. Poincaré ANL, 33 p. (2007, to appear)Google Scholar
  18. 18.
    Usami H. (1994) Nonexistence of positive entire solutions for elliptic inequalities of the mean curvature type. J. Differ. Equ. 111, 472–480MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yang Z. (2006) Existence of explosive positive solutions of quasilinear elliptic equations. Appl. Math. Comput. 177, 581–588MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Roberta Filippucci
    • 1
  • Patrizia Pucci
    • 1
  • Marco Rigoli
    • 2
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly

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