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On a class of semilinear fractional elliptic equations involving outside Dirac data

The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-Δ)^α u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(-Δ)^α +u^p} u=δ_{0}& \ {\rm in}\ \ \mathbb{R}^N\setminus B_1(e_N), \end{array} \end{equation} where $p>0$, $ (-Δ)^α$ with $α\in(0,1)$ denotes the fractional Laplacian operator in the principle value sense, $B_1(e_N)$ is the unit ball centered at $e_N=(0,\cdots,0,1)$ in $\mathbb{R}^N$ with $N\ge 2$ and $δ_0$ is the Dirac mass concentrated at the origin. We prove that problem (\ref{00}) admits a unique weak solution when $p> 1+\frac{2α}{N}$. Moreover, if in addition $p\ge \frac{N+2}{N-2}$, the weak solution vanishes as $α\to 1^-$. We also show that problem (\ref{00}) doesn't have any weak solution when $p\in[0, 1+\frac{2α}{N}]$. These results are very surprising since there are in total contradiction with the classical setting, i.e. $$ \arraycolsep=1pt \begin{array}{lll} -Δu+ u^p=0\ \ \ \ &\ {\rm in}\ \B_1(e_N),\\[2mm] \phantom{-Δ+u^{p} } u=δ_{0}& \ {\rm in}\ \ \R^N\setminus B_1(e_N), \end{array} $$ for which it has been proved that there are no solutions for $p\ge \frac{N+1}{N-1}$.