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.