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Liouville-type results for semilinear elliptic equations in unbounded domains

  • Henri BerestyckiEmail author
  • FranÇois Hamel
  • Luca Rossi
Article

Abstract

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.

Keywords

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

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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

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