Positive solutions of semilinear problems in an exterior domain of R2 $\mathbb{R}^{2}$

Abstract

Abstract The aim of this paper is to establish the existence and global asymptotic behavior of a positive continuous solution for the following semilinear problem: {−Δu(x)=a(x)uσ(x),x∈D,u>0,in D,u(x)=0,x∈∂D,lim|x|→∞u(x)ln|x|=0, $$ \textstyle\begin{cases} -\Delta u(x)=a(x)u^{\sigma }(x), \quad x\in D, \\ u>0, \quad \text{in }D, \\ u(x)=0, \quad x\in \partial D, \\ \lim_{ \vert x \vert \rightarrow \infty }\frac{u(x)}{ \ln \vert x \vert }=0, \end{cases} $$ where σ<1 $\sigma <1$, D is an unbounded domain in R2 $\mathbb{R}^{2}$ with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. As main tools, we use Kato class, Karamata regular variation theory and the Schauder fixed point theorem.

Keywords

• Singular positive solution
• Blow-up
• Green’s function
• Karamata class
• Kato class