TY - JOUR
T1 - A direct method of moving spheres on fractional order equations
AU - Chen, Wenxiong
AU - Li, Yan
AU - Zhang, Ruobing
N1 - Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/5/15
Y1 - 2017/5/15
N2 - In this paper, we introduce a direct method of moving spheres for the fractional Laplacian (−△)α/2 with 0>α>2, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify non-negative solutions for semilinear equations involving the fractional Laplacian in Rn; we prove a non-existence result for the prescribing Qα curvature equation on Sn; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on a half Euclidean space. We expect to see more applications of this method to many other nonlinear equations involving non-local operators.
AB - In this paper, we introduce a direct method of moving spheres for the fractional Laplacian (−△)α/2 with 0>α>2, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify non-negative solutions for semilinear equations involving the fractional Laplacian in Rn; we prove a non-existence result for the prescribing Qα curvature equation on Sn; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on a half Euclidean space. We expect to see more applications of this method to many other nonlinear equations involving non-local operators.
KW - Direct method of moving spheres
KW - Fractional Laplacians
KW - Non-existence of solutions
KW - Radial symmetry
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U2 - 10.1016/j.jfa.2017.02.022
DO - 10.1016/j.jfa.2017.02.022
M3 - Article
AN - SCOPUS:85014778371
SN - 0022-1236
VL - 272
SP - 4131
EP - 4157
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 10
ER -