Skip to main content
SpringerLink
Log in

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

  • Published:
Mathematical Notes Aims and scope Submit manuscript

An Erratum to this article was published on 01 April 2009

Abstract

In this paper, we consider the generalized solutions of the inequality

$$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$

on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality − Δu ≥ u q, q > 1, on spherically symmetric noncompact Riemannian manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Gidas and J. Spruck, “Global and local behavior of positive solutions of nonlinear elliptic equations,” Comm. Pure Appl. Math. 34(4), 525–598 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Grigor’yan, “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,” Bull. Amer. Math. Soc. 36(2), 135–249 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. G. Losev, “On some Liouville theorems on noncompact Riemannian manifolds,” Sibirsk. Mat. Zh. 39(1), 87–93 (1998) [Siberian Math. J. 39 (1), 74–80 (1998)].

    MATH  MathSciNet  Google Scholar 

  4. A. G. Losev and E. A. Mazepa, “Bounded solutions of the Schrödinger equation on Riemannian products,” Algebra Anal. 13(1), 84–110 (2001) [St. Petersbg. Math. J. 13 (1), 57–73 (2002)].

    MathSciNet  Google Scholar 

  5. V. M. Miklyukov and V. G. Tkachev, “Denjoy-Alfors’s theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces,” Comm. Anal. Geom. 4(4), 547–587 (1996).

    MATH  MathSciNet  Google Scholar 

  6. E. Mitidieri and S. I. Pokhozhaev, “The absence of global positive solutions of quasilinear elliptic inequalities,” Dokl. Ross. Akad. Nauk 359(4), 456–460 (1998) [Dokl. Math. 57 (2), 250–253 (1998)].

    MATH  MathSciNet  Google Scholar 

  7. J. Serrin and H. Zou, “Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,” Acta. Math. 189(1), 79–142 (2002).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Volgograd State University, Russia

    A. G. Losev & Yu. S. Fedorenko

Authors
  1. A. G. Losev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. S. Fedorenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © A. G. Losev, Yu. S. Fedorenko, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 6, pp. 867–878.

An erratum to this article can be found online at http://dx.doi.org/10.1134/S0001434609030183.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Losev, A.G., Fedorenko, Y.S. Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds. Math Notes 81, 778–787 (2007). https://doi.org/10.1134/S0001434607050252

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434607050252

Key words

Search

Navigation