Remark on some conformally invariant integral equations: the method of moving spheres

By Yanyan Li. 2003

439 citations recorded by the source corpus.

where a > 0 is some constant and x ∈ R. Hypothesis (2) was removed by Caffarelli, Gidas and Spruck in [8]; this is important for applications. Such Liouville type theorems have been extended to general conformally invariant fully nonlinear equations by Li and Li ([24]–[27]); see also related works of Viaclovsky ([40]–[41]) and Chang, Gursky and Yang ([13]–[14]). The method used in [21], as well as in much of the above cited work, is the method of moving planes. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations; see Aleksandrov [1], Serrin [38], Gidas, Ni and Nirenberg [21]–[22], Berestycki and Nirenberg [2], and others. In [30], Li and Zhu gave a proof of the above mentioned theorem of Caffarelli, Gidas and Spruck using the method of moving spheres (i.e. the method of moving planes together with the conformal invariance), which fully exploits the conformal invariance of the problem and, as a result, captures the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions. Significant simplifications to the proof in [30] have been made in Li and Zhang [29]. The method of moving spheres has been used in [24]–[27]. Liouville type theorems for various conformally invariant equations have received much attention; see, in addition to the above cited papers, [23], [17], [15], [33], [42] and [43].

Subfields: Applied Mathematics

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Citations in the corpus

On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, methodology): “12), while the regularity assumption u ∈ L∞loc(R ) was weakened by Li [73] to u ∈ L loc (R ).”
A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems (2007, neutral): “Then ‖vk‖∞ = 1 and either vk(0) = 1 for all k ∈ N (positive blow-up) or vk(0) = −1 for all k ∈ N (negative blow-up).”
On the fractional Lane-Emden equation (2014, result): “The aforementioned classiﬁcation results of Gidas-Spruck and Caﬀarelli-Gidas-Spruck have been generalized to the fractional setting (see Y. Li [24] and Chen-Li-Ou [8]).”
Delaunay-type singular solutions for the fractional Yamabe problem (2015, background): “13 has been studied by different authors if N > 2γ, even for 1 < p < N+2γ N−2γ (see [11, 10, 22]), but in this paper we need this result also for 2γ ≥ N since we have reduced our problem to dimension N = 1 for any γ ∈ (0, 1).”
An overdetermined problem in Riesz-potential and fractional Laplacian (2011, neutral): “[33]; see also [34] for the moving sphere method in integral forms, the book [35] and an exhaustive list of references therein, where the symmetry of solutions in the entire space was proved.”
A fractional Yamabe flow and some applications (2011, background): “(¯v)+ 1 1 −m v¯N φ = 0, i.e., ¯v solves (50). Finally, it follows from Theorem A.2 that v(·,s) converges to ¯v in C3(Sn). Proof of Theorem 1.3. By the classiﬁcation of solutions of (50) in [23] and [57], Th eorem 1.3 follows from Theorem 5.1 immediately. From Theorem 1.3 we see that the extinction ”
SHARP HARDY-LITTLEWOOD-SOBOLEV INEQUALITY ON THE UPPER HALF SPACE (2013, background): “Our results are closely related to the study of fractional Lapalacian operator in the whole space [8], fractional Yamabe problem [9, 22, 23, 25, 33, 34], fractional 6”
Uniqueness of Radial Solutions for the Fractional Laplacian (2013, methodology): “However, both the uniqueness proofs in [16, 29] make essential use of the fact that this problem exhibits conformal symmetry.”
Sharp reversed Hardy–Littlewood–Sobolev inequality on Rn (2015, methodology): “To realize that u ≡ v, we observe that u given in (4.25) satisfies the following equation u(x) = ∫ Rn |x− y|pu(y)−(1+2n)/pdy; see [Li04, Appendix A].”
Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R (2010, background): “• Optimizers for fractional Sobolev inequalities in d > 1 dimensions; see [12, 21].”