Singular solutions of a nonlinear elliptic equation in a punctured domain

Abstract

We consider the following semilinear problem \begin{equation*} \begin{cases} -\Delta u(x)=a(x)u^{\sigma }(x),\text{ }x\in \Omega \backslash \{0\}\text{ (in the distributional sense),} \\ u>0,\text{ in }\Omega \backslash \{0\},\\ \underset{\left\vert x\right\vert \rightarrow 0}{\lim }\left\vert x\right\vert ^{n-2}u(x)=0, \\ u(x)=0,\text{ }x\in \partial \Omega , \end{cases}% \end{equation*} where $\sigma <1,$ $\Omega $ is a bounded regular domain in $\mathbb{R}^{n}$ $(n\geq 3)$ containing $0$ and $a$ is a positive continuous function in $\Omega \backslash \{0\}$, which may be singular at $x=0$ and/or at the boundary $\partial \Omega$. When the weight function $a(x)$ satisfies suitable assumption related to Karamata class, we prove the existence of a positive continuous solution on $\overline{\Omega }\backslash \{0\}$, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Keywords

• singular positive solution
• green's function
• karamata class
• kato class
• blow-up