- Research Article
- Open Access
Quasi-concavity for semilinear elliptic equations with non-monotone and anisotropic nonlinearities
Boundary Value Problems volume 2006, Article number: 80347 (2006)
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.
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.
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
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
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
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
Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin; 1998.
Greco A, Lucia M: Gamma-star-shapedness for semilinear elliptic equations. Communications on Pure and Applied Analysis 2005,4(1):93–99.
Greco A, Reichel W: Existence and starshapedness for the Lane-Emden equation. Applicable Analysis. An International Journal 2001,78(1–2):21–32.
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
Kawohl B: Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics. Volume 1150. Springer, Berlin; 1985:iv+136.
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
Lewis JL: Capacitary functions in convex rings. Archive for Rational Mechanics and Analysis 1977,66(3):201–224.
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.
Protter MH, Weinberger HF: Maximum Principles in Differential Equations. Prentice-Hall, New Jersey; 1967:x+261.
Salani P: Starshapedness of level sets of solutions to elliptic PDEs. Abstract and Applied Analysis 2005,84(12):1185–1197.
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
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Equation
- Structural Condition