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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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


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