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abstract: |
    In this paper, we study the existence of nonnegative weak solutions to (E) $ (-\Delta)^\alpha
    u+h(u)=\nu $ in a general regular domain $\Omega$, which vanish in ${\mathbb{R}}^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a nonnegative Radon measure and $h:{\mathbb{R}}_+\to{\mathbb{R}}_+$ is a continuous nondecreasing function satisfying a subcritical integrability condition.

    Furthermore, we analyze properties of weak solution $u_k$ to $(E)$ with $\Omega={\mathbb{R}}^N$, $\nu=k\delta_0$ and $h(s)=s^p$, where $k>0$, $p\in(0,\frac{N}{N-2\alpha})$ and $\delta_0$ denotes Dirac mass at the origin. Finally, we show for $p\in(0,1+\frac{2\alpha}{N}]$ that $u_k\to\infty$ in ${\mathbb{R}}^N$ as $k\to\infty$, and for $p\in(1+\frac{2\alpha}{N},\frac{N}{N-2\alpha})$ that $\lim_{k\to\infty}u_k(x)=c|x|^{-\frac{2\alpha}{p-1}}$ with $c>0$, which is a classical solution of $ (-\Delta)^\alpha u+u^p=0$ in ${\mathbb{R}}^N\setminus\{0\}$.
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[**Semilinear fractional elliptic equations with measures in unbounded domain** ]{}

[: Fractional Laplacian, Radon measure, Dirac mass, Singularities.]{}

[: 35R11, 35J61, 35R06]{}

Introduction
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Let $\Omega$ be a regular domain (not necessary bounded) of ${\mathbb{R}}^N$ ($N\geq2$), $\alpha\in(0,1)$ and $d\omega(x)=\frac{dx}{1+|x|^{N+2\alpha}}$. Denote by $\mathfrak{M}^b(\Omega)$ the space of the Radon measures $\nu$ in $\Omega$ such that ${\|\nu\|}_{\mathfrak{M}^b(\Omega)}:=|\nu|(\Omega)<+\infty$ and by $\mathfrak{M}^b_+(\Omega)$ the nonnegative cone. The purpose of this paper is to study the existence of nonnegative weak solutions to semilinear fractional elliptic equations $$\label{eq1.1}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+h(u)=\nu\quad &{ \rm in}\quad\Omega,\\
  \phantom{   