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Self-generated interior blow-up solutions of fractional elliptic equation with absorption. (English) Zbl 1363.35368
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.
35R11 Fractional partial differential equations
35B44 Blow-up in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs