Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brezis, H., Kamin, S.: Sublinear elliptic equations in \(\mathbb{R}^n\). Manuscr. Math. 74(1), 87–106 (1992)
Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10(1), 55–64 (1986)
Chipot, M.: \(\ell \) goes to plus infinity. Birkhauser Advanced Text, Basel (2002)
Chipot, M.: \(\ell \) goes to to plus infinity: an update. J. KSIAM 18(2), 107–127 (2014)
Chipot, M.: Asymptotic issues for some partial differential equations. Imperial College Press, London (2016)
Chipot, M.: On the asymptotic behaviour of some problems of the calculus of variations. J. Elliptic Parabol. Equ. 1, 307–323 (2015)
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)
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)
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)
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)
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)
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)
Hess, P.: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differ. Equ. 6(8), 951–961 (1981)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chipot, M., Dávila, J. & del Pino, M. On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains. J. Fixed Point Theory Appl. 19, 205–213 (2017). https://doi.org/10.1007/s11784-016-0349-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-016-0349-1