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

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

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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