Differential and Integral Equations

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

Huyuan Chen, Patricio Felmer, and Alexander Quaas

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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

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

First available in Project Euclid: 23 June 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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