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

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.

Keywords

• Positive solutions
• asymptotic behavior
• Dirichlet problem
• subsolution
• supersolution