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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 4

### Boundary blow-up solutions of elliptic equations involving regional fractional Laplacian

Pages: 989 – 1004

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n4.a6

#### Authors

#### Abstract

In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian:\begin{gather}(-\Delta)^{\alpha}_{\Omega} u+f(u) = 0 & \textrm{in} & \Omega \\u = + \infty & \textrm{on} & \partial \Omega ,\end{gather}where $\Omega$ is a bounded open domain in $\mathbb{R}^N (N \geq 2)$ with $C^2$ boundary $\partial \Omega , \alpha \in (0,1)$ and the operator $(-\Delta)^\alpha_\Omega$ is the regional fractional Laplacian. When $f$ is a nondecreasing continuous function satisfying $f(0) \geq 0$ and some additional conditions, we address the existence and nonexistence of solutions for this problem. Moreover, we further analyze the asymptotic behavior of solutions to it.

#### Keywords

regional fractional Laplacian, boundary blow-up solution, asymptotic behavior

#### 2010 Mathematics Subject Classification

35B40, 35B44, 35J61

H. Chen is supported by NSFC (No:11726614, 11661045), by the Jiangxi Provincial Natural Science Foundation, No: 20161ACB20007, and by Doctoral Research Foundation of Jiangxi Normal University.

Received 2 June 2018

Accepted 24 February 2019

Published 25 October 2019