Electron. J. Differential Equations, Vol. 2018 (2018), No. 137, pp. 1-14.

Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in exterior domains

Habib Maagli, Abdulah Khamis Alzahrani, Zagharide Zine El Abidine

Abstract:
In this article, we study the existence, uniqueness and the asymptotic behavior of a positive classical solution to the semilinear boundary value problem
$$\displaylines{ -\Delta u=a(x)u^{\sigma }\quad \text{in }D, \cr u|_{\partial D}=0,\quad \lim_{|x|\to \infty}u(x) =0. }$$
Here D is an unbounded regular domain in $\mathbb{R} ^n$ ($n\geq 3$) with compact boundary, $\sigma<1$ and the function a is a nonnegative function in $C_{\rm loc}^{\gamma}(D)$, $0<\gamma<1$, satisfying an appropriate assumption related to Karamata regular variation theory.

Submitted September 20, 2017. Published July 1, 2018.
Math Subject Classifications: 34B16, 34B18, 35B09, 35B40.
Key Words: Positive solutions; asymptotic behavior; Dirichlet problem; subsolution; supersolution.

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Habib Mâagli
King Abdulaziz University
College of Sciences and Arts, Rabigh Campus
Department of Mathematics. P. O. Box 344
Rabigh 21911, Saudi Arabia
email: habib.maagli@fst.rnu.tn
  Abdulah Khamis Alzahrani
King Abdulaziz University, Faculty of Sciences
Department of Mathematics. P. O. Box 80203
Jeddah 21589, Saudi Arabia
email: akalzahrani@kau.edu.sa
Zagharide Zine El Abidine
Université de Tunis El Manar
Faculté des Sciences de Tunis
UR11ES22 Potentiels et Probabilités
2092 Tunis, Tunisie
email: Zagharide.Zinelabidine@ipeib.rnu.tn

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