Skip to main content
  • Research Article
  • Open access
  • Published:

Quasi-concavity for semilinear elliptic equations with non-monotone and anisotropic nonlinearities

Abstract

A boundary-value problem for a semilinear elliptic equation in a convex ring is considered. Under suitable structural conditions, any classical solution lying between its (constant) boundary values is shown to decrease along each ray starting from the origin, and to have convex level surfaces.

[123456789101112131415]

References

  1. Acker A: On the uniqueness, monotonicity, starlikeness, and convexity of solutions for a nonlinear boundary value problem in elliptic PDEs. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1994,22(6):697–705.

    Article  MathSciNet  MATH  Google Scholar 

  2. Caffarelli LA, Spruck J: Convexity properties of solutions to some classical variational problems. Communications in Partial Differential Equations 1982,7(11):1337–1379. 10.1080/03605308208820254

    Article  MathSciNet  MATH  Google Scholar 

  3. Colesanti A, Salani P: Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations. Mathematische Nachrichten 2003,258(1):3–15. 10.1002/mana.200310083

    Article  MathSciNet  MATH  Google Scholar 

  4. Diaz JI, Kawohl B: On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings. Journal of Mathematical Analysis and Applications 1993,177(1):263–286. 10.1006/jmaa.1993.1257

    Article  MathSciNet  MATH  Google Scholar 

  5. Gabriel RM: A result concerning convex level surfaces of-dimensional harmonic functions. Journal of the London Mathematical Society. Second Series 1957, 32: 286–294. 10.1112/jlms/s1-32.3.286

    Article  MathSciNet  MATH  Google Scholar 

  6. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin; 1998.

    MATH  Google Scholar 

  7. Greco A, Lucia M: Gamma-star-shapedness for semilinear elliptic equations. Communications on Pure and Applied Analysis 2005,4(1):93–99.

    Article  MathSciNet  MATH  Google Scholar 

  8. Greco A, Reichel W: Existence and starshapedness for the Lane-Emden equation. Applicable Analysis. An International Journal 2001,78(1–2):21–32.

    Article  MathSciNet  MATH  Google Scholar 

  9. Grossi M, Molle R: On the shape of the solutions of some semilinear elliptic problems. Communications in Contemporary Mathematics 2003,5(1):85–99. 10.1142/S0219199703000914

    Article  MathSciNet  MATH  Google Scholar 

  10. Kawohl B: Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics. Volume 1150. Springer, Berlin; 1985:iv+136.

    Google Scholar 

  11. Korevaar NJ: Convexity of level sets for solutions to elliptic ring problems. Communications in Partial Differential Equations 1990,15(4):541–556. 10.1080/03605309908820698

    Article  MathSciNet  MATH  Google Scholar 

  12. Lewis JL: Capacitary functions in convex rings. Archive for Rational Mechanics and Analysis 1977,66(3):201–224.

    Article  MathSciNet  MATH  Google Scholar 

  13. Longinetti M: A maximum principle for the starshape of solutions of nonlinear Poisson equations. Unione Matematica Italiana. Bollettino. A. Serie VI 1985,4(1):91–96.

    MathSciNet  MATH  Google Scholar 

  14. Protter MH, Weinberger HF: Maximum Principles in Differential Equations. Prentice-Hall, New Jersey; 1967:x+261.

    MATH  Google Scholar 

  15. Salani P: Starshapedness of level sets of solutions to elliptic PDEs. Abstract and Applied Analysis 2005,84(12):1185–1197.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Greco.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Greco, A. Quasi-concavity for semilinear elliptic equations with non-monotone and anisotropic nonlinearities. Bound Value Probl 2006, 80347 (2006). https://doi.org/10.1155/BVP/2006/80347

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/BVP/2006/80347

Keywords