# Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$ R N

Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional... Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent: \begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned} ( - Δ ) σ u = K ( y ) u N + 2 σ N - 2 σ in R N , u > 0 , y ∈ R N . For $$N\ge 4$$ N ≥ 4 and $$\sigma \in (\frac{1}{2}, 1),$$ σ ∈ ( 1 2 , 1 ) , we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$ R N

, Volume 56 (4) – Jul 14, 2017
41 pages

/lp/springer_journal/local-uniqueness-and-periodicity-for-the-prescribed-scalar-curvature-70F8bGnh1q
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1194-9
Publisher site
See Article on Publisher Site

### Abstract

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent: \begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned} ( - Δ ) σ u = K ( y ) u N + 2 σ N - 2 σ in R N , u > 0 , y ∈ R N . For $$N\ge 4$$ N ≥ 4 and $$\sigma \in (\frac{1}{2}, 1),$$ σ ∈ ( 1 2 , 1 ) , we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 14, 2017

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