Radial symmetry for systems of fractional Laplacian. (English) Zbl 1438.35082
Summary: In this paper, we consider systems of fractional Laplacian equations in \(\mathbb{R}^n\) with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions \(u_i\) in the critical cases by using a direct method of moving planes introduced in [W. Chen et al., Adv. Math. 308, 404–437 (2017; Zbl 1362.35320)] and some new maximum principles in [C. Li et al. , “Maximum principles and Bôcher type theorems”, Proc. Natl. Acad. Sci. USA 115, No. 27, 6976–6979 (2018; doi:10.1073/pnas.1804225115].
MSC:
| 35B50 | Maximum principles in context of PDEs |
| 35R11 | Fractional partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
system of fractional Laplacian; method of moving planes; maximum principles with singular point; Kelvin transformCitations:
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