Self-generated interior blow-up solutions of fractional elliptic equation with absorption. (English) Zbl 1363.35368

Chen, Huyuan; Felmer, Patricio; Quaas, Alexander

Differ. Integral Equ. 28, No. 9-10, 839-860 (2015).

Summary: In this paper, we study positive solutions to problems involving the fractional Laplacian \[ \begin{cases} (-\Delta)^\alpha u(x)+|u|^{p-1}u(x)=0,& x\in\Omega\backslash\mathcal C,\\ u(x)=0,& x\in\Omega^c,\\ \lim_{x\in \Omega\backslash\mathcal C,\;x\to\mathcal C}u(x)=+\infty,\end{cases}\tag{0.1} \] where \(p>1\) and \(\Omega\) is an open bounded \(C^2\) domain in \(\mathbb{R}^N\), \(C\subset\Omega\) 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.

MSC:

35R11	Fractional partial differential equations
35B44	Blow-up in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

fractional Laplacian; asymptotic behavior; blow-up; positive solution