Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 95, pp. 1-14.

Title: Asymptotic behavior of positive solutions
of a semilinear Dirichlet problem  outside  the unit ball
        
Authors: Habib Maagli (King Abdulaziz Univ., Rabigh, Saudi Arabia)
         Sameh Turki (Faculte des Sciences de Tunis, Tunisia)
         Zagharide Zine El Abidine (Faculte des Sciences de Tunis, Tunisia)

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

Submitted February 7, 2013. Published April 11, 2013.
Math Subject Classifications: 31C35, 34B16, 60J50.
Key Words: Asymptotic behavior; Dirichlet problem; subsolution; supersolution.