On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains



The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.


Semilinear elliptic equation Cylindrical domain Asymptotic behavior 

Mathematics Subject Classification

35J15 35J25 35J61 



This work has been performed during a visit of the first author at the Universidad de Chile in Santiago and at the SBAI at the Sapienza Università di Roma. He would like to thank both institutions for their kind hospitality. The second and third authors have been supported by Grants Fondecyt 1130360, 1150066, Fondo Basal CMM and Millenium Nucleus CAPDE NC130017.


  1. 1.
    Brezis, H., Kamin, S.: Sublinear elliptic equations in \(\mathbb{R}^n\). Manuscr. Math. 74(1), 87–106 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10(1), 55–64 (1986)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chipot, M.: \(\ell \) goes to plus infinity. Birkhauser Advanced Text, Basel (2002)CrossRefMATHGoogle Scholar
  4. 4.
    Chipot, M.: \(\ell \) goes to to plus infinity: an update. J. KSIAM 18(2), 107–127 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Chipot, M.: Asymptotic issues for some partial differential equations. Imperial College Press, London (2016)CrossRefMATHGoogle Scholar
  6. 6.
    Chipot, M.: On the asymptotic behaviour of some problems of the calculus of variations. J. Elliptic Parabol. Equ. 1, 307–323 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chipot, M., Mojsic, A., Roy, P.: On some variational problems set on domains tending to infinity. Discrete Contin. Dyn. Syst. Ser. A 36(7), 3603–3621 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chipot, M., Rougirel, A.: On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded. Commun. Contemp. Math. 4, 15–44 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cinti, E., Dávila, J., del Pino, M.: Solutions of the fractional Allen–Cahn equation which are invariant under screw motion. J. Lond. Math. Soc (to appear)Google Scholar
  10. 10.
    Clément, P., Sweers, S.: Existence and multiplicity results for a semilinear elliptic eigenvalue problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14(1), 97–121 (1987)MathSciNetMATHGoogle Scholar
  11. 11.
    de Figueiro, D.: On the uniqueness of positive solutions of the Dirichlet problem \(-\Delta u=\lambda \sin (u)\). Nonlinear partial differential equations and their applications. College de France seminar, Vol. VII (Paris, 1983, 1984), 4, 80–83, Res. Notes Math., 122, Pitman, Boston, MA (1985)Google Scholar
  12. 12.
    del Pino, M., Musso, M., Pacard, F.: Solutions of the Allen–Cahn equation which are invariant under screw-motion. Manuscr. Math. 138(3–4), 273–286 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hess, P.: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differ. Equ. 6(8), 951–961 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institute of Applied MathematicsUniversity of ZürichZürichSwitzerland
  2. 2.Departamento de Ingenierıa Matemática and Centro de Modelamiento MatemáticoUniversidad de ChileSantiagoChile

Personalised recommendations