Skip to main content
SpringerLink
Log in

Pointwise estimates of solutions to semilinear elliptic equations and inequalities

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We obtain sharp pointwise estimates for positive solutions to the equation −Lu + Vuq = f, where L is an elliptic operator in divergence form, q ∈ ℝ\{0}, f ≥ 0 and V is a function that may change sign, in a domain Ω in ℝn, or in a weighted Riemannian manifold.

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. A. Agmon, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in Methods of Functional Analysis and Theory of Elliptic Equations (Naples, 1982), Liguori, Naples, 1983, pp. 19–52.

    Google Scholar 

  2. M. Aizenman and B. Simon, Brownianmotion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209–273.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monographs in Math., Springer-Verlag, London–Berlin–Heidelberg, 2001.

    Book  MATH  Google Scholar 

  4. R. Atar, S. Athreya and Z.-Q. Chen, Exit time, Green function and semilinear elliptic equations, Electr. J. Probab. 14 (2009), 50–71.

    MATH  Google Scholar 

  5. H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. Ser. 8 1-B (1998), 223–262.

    Google Scholar 

  6. H. Brezis and S. Kamin, Sublinear elliptic equations in Rn, Manuscripta Math. 74 (1992), 87–106.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55–64.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. T. Cao and I. E. Verbitsky, Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms, Calc. Var. Partial Differential Equations 52 (2015), 529–546.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. T. Cao and I. E. Verbitsky, Nonlinear elliptic equations and intrinsic potentials of Wolff type, J. Func. Anal. 272 (2017), 112–165.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger’s Equation, Grundlehren Math. Wiss., Vol. 312, Springer, Berin–Heidelberg, 1995.

  11. J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Classics in Math., Springer-Verlag, New York–Berlin–Heidelberg–Tokyo, 2001.

    Book  MATH  Google Scholar 

  12. E. B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, University Lecture Ser. Vol. 34, Amer. Math. Soc., Providence, RI, 2004.

    Book  MATH  Google Scholar 

  13. L. Dupaigne, M. Ghergu, and V. Radulescu, Lane–Emden–Fowler equations with convection and singular potential, J. Math. Pures Appl. 87 (2007), 563–581.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Frazier and I. E. Verbitsky, Global Green’s function estimates, in Around the Research of Vladimir Maz’ya III, Analysis and Applications, Intern. Math. Series, Vol. 13, Springer, Berelin, 2010, pp. 105–152.

    Chapter  Google Scholar 

  15. M. Frazier, F. Nazarov, and I. E. Verbitsky, Global estimates for kernels of Neumann series and Green’s functions, J. London Math. Soc. 90 (2014), 903–918.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Ghergu, Lane–Emden systems with negative exponents, J. Funct. Anal. 258 (2010), 3295–3318.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Ghergu and V. Radulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. in Math. and Appl., Vol. 37, Oxford University Press, Oxford, 2008.

  18. A. Grigor’yan, Heat kernels onweighted manifolds and applications, Contemp.Math. 398 (2006), 93–191.

    Article  MATH  Google Scholar 

  19. A. Grigor’yan, Heat Kernel and Analysis on Manifolds, AMS/IP Studies in Adv. Math., Vol. 4, Amer. Math. Soc., Providence, RI, 2009.

    MATH  Google Scholar 

  20. A. Grigor’yan and W. Hansen, Lower estimates for a perturbed Green function, J. Anal. Math. 104 (2008), 25–58.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Grigor’yan and W. Hansen, Lower estimates for perturbed Dirichlet solutions, unpublished manuscript, https://www.math.uni-bielefeld.de/˜grigor/fekac.pdf

  22. W. Hansen, Uniform boundary Harnack principle and generalized triangle property, J. Funct. Anal. 226 (2005), 452–484.

    Article  MathSciNet  MATH  Google Scholar 

  23. W. Hansen and Z. Ma, Perturbation by differences of unbounded potentials, Math. Ann. 287 (1990), 553–569.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Hansen and I. Netuka, On the Picard principle for + μ, Math. Z. 270 (2012), 783–807.

    Article  MathSciNet  MATH  Google Scholar 

  25. I. W. Herbst and Z. Zhao, Green’s functions for the Schrödinger equation with short-range potential, Duke Math. J. 59 (1989), 475–519.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. J. Jaye, V. G. Maz’ya and I. E. Verbitsky, Existence and regularity of positive solutions of elliptic equations of Schrödinger type, J. Anal.Math. 118 (2012), 577–621.

    Article  MathSciNet  MATH  Google Scholar 

  27. N. J. Kalton and I. E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. 351 (1999), 3441–3497.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Ser. Nonlin. Analysis Appl., Vol. 21, Berlin–Boston, 2014.

  29. M. Murata, Structure of positive solutions to (− + V)u = 0 in Rn, Duke Math. J. 53 (1986), 869–943.

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Pinchover, Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations, in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proc. Symp. PureMath. 76, Amer.Math. Soc., Providence, RI, 2007, pp. 329–356.

    Google Scholar 

  31. L. Véron, Elliptic equations involving measures, in Stationary Partial Differential Equations, Stationary Partial Differential Equations, Handbook of Differential Equations, Vol. 1, Elsevier, Amsterdam, 2004, pp. 593–712.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bielefeld, 33501, Bielefeld, Germany

    Alexander Grigor’yan

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Igor Verbitsky

Authors
  1. Alexander Grigor’yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Verbitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Grigor’yan.

Additional information

The first author is supported in part by SFB 701 of the German Research Council.

The second author is supported in part by NSF grant DMS-1161622.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grigor’yan, A., Verbitsky, I. Pointwise estimates of solutions to semilinear elliptic equations and inequalities. JAMA 137, 559–601 (2019). https://doi.org/10.1007/s11854-019-0004-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-019-0004-z

Search

Navigation