Liouville theorem and isolated singularity of fractional Laplacian system with critical exponents. (English) Zbl 1430.35040
Summary: This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving spheres to derive a Liouville Theorem with at most three radial solutions, and then prove the solutions in \(\mathbb{R}^n \backslash \{0\}\) are radially symmetric and monotonically decreasing. Together with blow up analysis, we get the upper bound of the local solutions in \(B_1 \backslash \{0\}\). Our results is an extension of the classical works by L. A. Caffarelli et al. [Commun. Pure Appl. Math. 42, No. 3, 271–297 (1989; Zbl 0702.35085); Arch. Ration. Mech. Anal. 213, No. 1, 245–268 (2014; Zbl 1296.35208)], Z. Chen and C.-S. Lin [Math. Ann. 363, No. 1–2, 501–523 (2015; Zbl 1337.35046)] and Y. Guo and J. Liu [Commun. Partial Differ. Equations 33, No. 2, 263–284 (2008; Zbl 1139.35305)].
MSC:
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35B33 | Critical exponents in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35R11 | Fractional partial differential equations |
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