Semilinear fractional elliptic equations with measures in unbounded domain

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 ${\mathbb{R}}^N\setminus \{0\}$.

30 pages