Revista Matemática Iberoamericana

Some nonexistence results for positive solutions of elliptic equations in unbounded domains

Lucio Damascelli and Francesca Gladiali

Full-text: Open access

Abstract

We prove some Liouville type theorems for positive solutions of semilinear elliptic equations in the whole space $\mathbb{R}^N$, $N\geq 3$, and in the half space $\mathbb{R}^N_{+}$ with different boundary conditions, using the technique based on the Kelvin transform and the Alexandrov-Serrin method of moving hyperplanes. In particular we get new nonexistence results for elliptic problems in half spaces satisfying mixed (Dirichlet-Neumann) boundary conditions.

Article information

Source
Rev. Mat. Iberoamericana Volume 20, Number 1 (2004), 67-86.

Dates
First available in Project Euclid: 2 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.rmi/1080928420

Zentralblatt MATH identifier
02104138

Mathematical Reviews number (MathSciNet)
MR2076772

Subjects
Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B45: A priori estimates 35B50: Maximum principles

Keywords
Liouville theorems Kelvin transform maximum principle moving plane

Citation

Damascelli, Lucio; Gladiali, Francesca. Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Revista Matemática Iberoamericana 20 (2004), no. 1, 67--86. http://projecteuclid.org/euclid.rmi/1080928420.


Export citation

References

  • Almeida, L., Damascelli, L., Ge, Y.: A few symmetry results for nonlinear elliptic PDE on noncompact manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 3, 313-342.
  • Bianchi, G.: Non-existence of positive solutions to semilinear elliptic equations on $\mathbbR^N$ or $\mathbbR^N_+$ through the method of moving planes. Comm. Partial Differential Equations 22 (1997), no. 9 10, 1671-1690.
  • Berestycki, H., Caffarelli, L. A. and Nirenberg, L.: Symmetry for elliptic equations in a half space. In Boundary value problems for partial differential equations and applications, 27-42. RMA Res. Notes Appl. Math. 29. Masson, Paris 1993.
  • Berestycki, H., Caffarelli, L. A. and Nirenberg, L.: Inequalities for second order elliptic equations with applications to unbounded domains I. Duke Math J. 81 (1996), 467-494.
  • Berestycki, H., Caffarelli, L. A. and Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipshitz domains. Comm. Pure Appl. Math. 50 (1997), 1089-1111.
  • Berestycki, H., Caffarelli, L. A. and Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), 69-94.
  • Berestycki, H., Grossi, M. and Pacella, F.: A nonexistence theorem for an equation with critical Sobolev exponent in the half space. Manuscripta Math. 77 (1992), 265-281.
  • Caffarelli, L. A., Gidas, B. and Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), 271-297.
  • Chen, C. C. and Lin, C. S.: Local behavior of singular positive solutions of semilinear elliptic equations with sobolev exponent. Duke Math. J. 78 (1995), no. 2, 315-334.
  • Chen, W. and Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), no. 3, 615-622.
  • Colorado Heras, E. and Peral Alonso, I.: Semilinear elliptic problems with mixed Dirchlet-Neumann boundary conditions. J. Funct. Anal. 199 (2003), no. 2, 468-507.
  • Dancer, E. N.: Some notes on the method of moving planes. Bull. Austral. Math. Soc. 46 (1992), 425-434.
  • Damascelli, L., Ramaswamy, M.: Symmetry of $C^1$ solutions of $p-$Laplace equations in $\mathbbR^N$. Adv. Nonlinear Stud. 1 (2001), no. 1, 40-64.
  • Gidas, B., Ni, W. M. and Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$. In Mathematical analysis and applications, Part A, 369-402. Adv. in Math. Suppl. Stud. 7a. Academic Press, New York-London, 1981.
  • Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), 525-598.
  • Gidas, B., Spruck, J.: A Priori Bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), 883-901.
  • Lou, Y., Zhu, M.: Classification of nonnegative solutions to some elliptic problems. Differential Integral Equations 12 (1999), no. 4, 601-612.
  • Pucci, P., Serrin, J. and Zou, H.: A strong maximum principle for quasilinear elliptic equations. J. Math. Pures Appl. (9) 78 (1999), 769-789.
  • Protter, M. H., Weinberger, H. F.: Maximum Principle in Differential Equations. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967.
  • Terracini, S.: Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions. Differential Integral Equations 8 (1995), 1911-1922.
  • Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differential Equations 1 (1996), no. 2, 241-264.
  • Vázquez, J. L.: A Strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191-202.