Liouville-type results for semilinear elliptic equations in unbounded domains

  • Henri BerestyckiEmail author
  • FranÇois Hamel
  • Luca Rossi


This paper is devoted to the study of some class of semilinear elliptic equations in the whole space:
$$ -a_{ij}(x)\partial_{ij}u(x) -- q_i(x)\partial_iu(x)=f(x,u(x)),\quad x\in{\mathbb R}^N. $$
The aim is to prove uniqueness of positive- bounded solutions—Liouville-type theorems. Along the way, we establish also various existence results.

We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u= 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one. We also derive the large time behavior for the associated evolution equation.


Linear and semi-linear elliptic equations Principal eigenvalues Maximum principles Liouville-type results Periodic and almost-periodic equations 


  1. 1.
    Berestycki, H.: Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Funct. Anal. 40, 1–29 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berestycki, H., Dickmann, O., Nagelkerke, K., Zegeling, P.: Can a species keep pace with a shiftering? PreprintGoogle Scholar
  3. 3.
    Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. I: periodic framework. J. Eur. Math. Soc. 7, 173–213 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: I – influence of periodic heterogeneous environment on species persistence. J. Math. Biol. 51, 75–113 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berestycki, H., Lions, P.-L.: Some applications of the method of super- and sub-solutions. In: Bardos, C., Lasry, J.M., Schatzman, M. (eds.) Bifurcation and Nonlinear Eigenvalue Problems. Lecture Notes in Mathematics, vol. 782, pp. 16–41. Springer-Verlag, New York (1980)Google Scholar
  6. 6.
    Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 497–572 (1992)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Berestycki, H., Nirenberg, L., Varadhan, S.R.S.: The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Commun. Pure Appl. Math. 47, 47–92 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. U.S.A. 48, 2039–2043 (1963)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Engländer, J., Kyprianou, A.E.: Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32, 78–99 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Engländer, J., Pinsky, R.G.: On the construction and support properties of measure-valued diffusions on \(D\subseteq R^d\) with spatially dependent branching. Ann. Probab. 27, 684–730 (1999)Google Scholar
  11. 11.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer-Verlag, Berlin (1983)zbMATHGoogle Scholar
  12. 12.
    Kolmogorov, A.N., Petrovskii, I.G., Piskunov, N.S.: Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. d'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), Sér. Int. A 1, 1–26 (1937)Google Scholar
  13. 13.
    Pinsky, R.G.: Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24, 237–267 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1967)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Henri Berestycki
    • 1
    Email author
  • FranÇois Hamel
    • 2
  • Luca Rossi
    • 1
    • 3
  1. 1.EHESS, CAMS54 Boulevard RaspailParisFrance
  2. 2.Université Paul Cézanne Aix-Marseille III, LATP (UMR CNRS 6632), F.S.T., Avenue Escadrille Normandie-NiemenMarseilleFrance
  3. 3.Dipartimento di MatematicaUniversità La Sapienza Roma IRomaItaly

Personalised recommendations