SINGULAR SOLUTIONS OF A FRACTIONAL DIRICHLET PROBLEM IN A PUNCTURED DOMAIN

Abstract

Let $D$ be a bounded regular domain in $\R^n\ (n \geq 3)$ containing $0$, $0< \alpha<2, $ and $\sigma < 1$. We take up in this article the existence and asymptotic behavior of a positive continuous solution for the following semi-linear fractional differential equation $$
(-\Delta)^{\frac{\alpha}{2}} u= a(x) u^{\sigma}(x) \ \ \text{in } \ D \backslash \{0\},$$
with the boundary Dirichlet condition
$$ \lim\limits_{|x| \to 0} |x|^{n-\alpha} u(x) =0\ \text{and}\ \lim\limits_{x \to \partial D} \delta(x)^{2-\alpha} u(x) =0,$$
where $\delta(x)= \text{dist}\ ( \ x,\ \partial D) $ denotes the Euclidean distance between $x $ and $\partial D$ and the function $a$ is a positive continuous function in $D \backslash \{0\}$, which may be singular at
$x = 0$ and/or at the boundary $\partial D$ satisfying some appropriate assumption related to Karamata class. More precisely, we shall prove existence and global asymptotic behavior of a positive continuous solution on $\Bar{D}\backslash\{0\}$. We will use some potential theory arguments.