## Differential and Integral Equations

- Differential Integral Equations
- Volume 28, Number 9/10 (2015), 839-860.

### Self-generated interior blow-up solutions of fractional elliptic equation with absorption

Huyuan Chen, Patricio Felmer, and Alexander Quaas

#### Abstract

In this paper, we study positive solutions to problems involving the fractional Laplacian \begin{equation} \begin{cases} (-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=0,\ \ \ \ & x\in\Omega\setminus\mathcal{C},\\ \ \quad \ u(x)=0,\ & x\in\Omega^c,\\ \displaystyle \lim_{x\in\Omega\setminus\mathcal{C}, \ x\to\mathcal{C}}u(x)=+\infty, \end{cases} \tag*{(0.1)} \end{equation} where $p>1$ and $\Omega$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $\mathcal{C}\subset \Omega$ is a compact $C^2$ manifold with $N-1$ multiples dimensions and without boundary, the operator $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ is the fractional Laplacian. We consider the existence of positive solutions for problem (0.1). Moreover, we further analyze uniqueness, asymptotic behavior and nonexistence.

#### Article information

**Source**

Differential Integral Equations, Volume 28, Number 9/10 (2015), 839-860.

**Dates**

First available in Project Euclid: 23 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1435064542

**Mathematical Reviews number (MathSciNet)**

MR3360722

**Zentralblatt MATH identifier**

1363.35368

**Subjects**

Primary: 35R11: Fractional partial differential equations 35B44: Blow-up 35B40: Asymptotic behavior of solutions

#### Citation

Chen, Huyuan; Felmer, Patricio; Quaas, Alexander. Self-generated interior blow-up solutions of fractional elliptic equation with absorption. Differential Integral Equations 28 (2015), no. 9/10, 839--860. https://projecteuclid.org/euclid.die/1435064542