Semilinear fractional elliptic equations with measures in unbounded domain

Abstract

In this paper, we study the existence of nonnegative weak solutions to (E) $ (-Δ)^αu+h(u)=ν$ in a general regular domain $Ω$, which vanish in $\R^N\setminusΩ$, where $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$, $ν$ 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 $Ω=\mathbb{R}^N$, $ν=kδ_0$ and $h(s)=s^p$, where $k>0$, $p\in(0,\frac{N}{N-2α})$ and $δ_0$ denotes Dirac mass at the origin. Finally, we show for $p\in(0,1+\frac{2α}{N}]$ that $u_k\to\infty$ in $\mathbb{R}^N$ as $k\to\infty$, and for $p\in(1+\frac{2α}{N},\frac{N}{N-2α})$ that $\lim_{k\to\infty}u_k(x)=c|x|^{-\frac{2α}{p-1}}$ with $c>0$, which is a classical solution of $ (-Δ)^αu+u^p=0$ in $\mathbb{R}^N\setminus\{0\}$.

30 pages