Skip to main content
Log in

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

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

Abstract

We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.

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.

Similar content being viewed by others

References

  1. Birindelli, I., Galise, G., Ishii, H.: A family of degenerate elliptic operators: maximum principle and its consequences. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(2), 417–441 (2018)

    Article  MathSciNet  Google Scholar 

  2. Birindelli, I., Galise, G., Leoni, F.: Liouville theorems for a family of very degenerate elliptic nonlinear operators. Nonlinear Anal. 161, 198–211 (2017)

    Article  MathSciNet  Google Scholar 

  3. Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations, I. J. Fixed Point Theory Appl. 5(2), 353–395 (2009)

    Article  MathSciNet  Google Scholar 

  4. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  5. Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193–222 (1977)

    Article  MathSciNet  Google Scholar 

  6. del Pino, M.A.: A global estimate for the gradient in a singular elliptic boundary value problem. Proc. R. Soc. Edinb. Sect. A 122(3–4), 341–352 (1992)

    Article  MathSciNet  Google Scholar 

  7. Dolcetta, I.C., Leoni, F., Vitolo, A.: On the inequality \(F(x, D^2u)\ge f(u)+g(u)|Du|^q\). Math. Ann. 365(1–2), 423–448 (2016)

    Article  MathSciNet  Google Scholar 

  8. Felmer, P., Quaas, A., Sirakov, B.: Existence and regularity results for fully nonlinear equations with singularities. Math. Ann. 354(1), 377–400 (2012)

    Article  MathSciNet  Google Scholar 

  9. Galise, G.: On positive solutions of fully nonlinear degenerate Lane–Emden type equations. J. Differ. Equ. 266(2–3), 1675–1697 (2019)

    Article  MathSciNet  Google Scholar 

  10. Harvey, F.R., Lawson Jr., H.B.: Dirichlet duality and the nonlinear Dirichlet problem. Commun. Pure Appl. Math. 62(3), 396–443 (2009)

    Article  MathSciNet  Google Scholar 

  11. Harvey, F.R., Lawson Jr., H.B.: \(p\)-convexity, \(p\)-plurisubharmonicity and the Levi problem. Indiana Univ. Math. J. 62(1), 149–169 (2013)

    Article  MathSciNet  Google Scholar 

  12. Hernandez, J., Mancebo, F.J.: Singular elliptic and parabolic equations. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, pp. 317–400. Elsevier, Amsterdam (2006)

    Chapter  Google Scholar 

  13. Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991)

    Article  MathSciNet  Google Scholar 

  14. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Giulio Galise.

Additional information

Communicated by M. Del Pino.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Birindelli, I., Galise, G. The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity. Calc. Var. 58, 180 (2019). https://doi.org/10.1007/s00526-019-1629-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1629-6

Mathematics Subject Classification

Navigation