Skip to main content
Log in

Partial regularity of stable solutions to the Emden equation

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

An Erratum to this article was published on 17 November 2012

Abstract

We prove that for stable solutions of −Δu = e u, the dimension of their singular sets do not exceed n − 10.

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

Access this article

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. Brezis H., Merle F.: Uniform estimates and blow-up behavior for solutions of −Δu = V(x)e u in two dimensions. Commun. Partial Differ. Equ. 16, 1223–1253 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Crandall M.G., Rabinowitz P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58, 207–218 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Da Lio F.: Partial regularity for stationary solutions to Liouville-type equation in dimension 3. Commun. Partial Differ. Equ. 33(10), 1890–1910 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dancer E.N.: Finite Morse index solutions of exponential problems. Ann. Inst. H. Poincare Anal. Non Linaire 25, 173–179 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Davila, J., Dupaigne, L., Farina, A.: Partial regularity of finite Morse index solutions to the Lane–Emden equations. Preprint

  6. Farina A.: Stable solutions of −Δu = e u on \({\mathbb{R}^n}\) . Comptes Rendus Math. 345(2), 63–66 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)

    Google Scholar 

  8. Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49, 241–269 (1972/1973)

    Google Scholar 

  9. Lin, F.H., Yang, X.P.: Geometric Measure Theory: An Introduction. Advanced Mathematics (Beijing/Boston), vol. 1. Science Press/International Press, Boston/Beijing (2002)

  10. Pacard F.: Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: the supercritical case. Annales de l’Institut Henri Poincaré. Anal. Non linéaire 11(5), 537–551 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Wang, K.: Partial regularity of stable solutions to the supercritical equations and its applications. Preprint

  12. Xavier, C.: Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems. in: Perspectives in Nonlinear Partial Differential Equations: In Honor of Haim Brezis. Contemporary Mathematics, vol. 446. American Mathematical Society, Providence, RI (2007)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kelei Wang.

Additional information

Communicated by A. Malchiodi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, K. Partial regularity of stable solutions to the Emden equation. Calc. Var. 44, 601–610 (2012). https://doi.org/10.1007/s00526-011-0446-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-011-0446-3

Mathematics Subject Classification (2000)

Navigation