On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions

By Tianling Jin, Yanyan Li, Jingang Xiong. 2011

295 citations recorded by the source corpus.

We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/98ffc39a40ea5ec3bd9f8c07324be6ebd977029c

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Unique Continuation for Fractional Orders of Elliptic Equations (2016, background): “on U, while the function of interest is u, deﬁned on a null-set in Rn. Thus one has to use blow-up in spaces tighter than L∞. This is achieved through compactness of degenerate elliptic equations [11][18]. This leads to a lot of works involving fractional-order elliptic equations, see for instance [”
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space (2015, background): “4 in[22]) in these cubes we infer sup Q1 v(m) ≤ C1 inf Q1 v(m) ≤ C1.”
Direct Method of Moving Spheres on Fractional Order Equations (2015, background): “7) does not admit any positive solution. For the precise deﬁnition of Pα, please see section 4. In particular, it is the conformal Laplacian when α= 2 and the Penitz operator for α= 4. Remark 1.1. In [JLX1], Jin, Li, and Xiong obtained a necessary condition–a Kazdan-Warner type identityfor (1.7) to ”
The Nirenberg problem and its generalizations: a unified approach (2014, methodology): “e blow up point of vi, then vi has to have at least another blow up point. Proof. The proof is by checking the “balance” condition of Kazdan-Warner type (10). It is the same as that of Theorem 5.3 in [73]. Theorem 3.6. Suppose that K ∈ C1(Sn) (K ∈ C1,1(Sn) if 0 < σ ≤ 1/2), for some constant A1 &g”
A direct method of moving planes for the fractional Laplacian (2014, methodology): “larity assumptions on u. Note. After this paper has been posted on the arXiv, Xiong brought to our attention that they proved a maximum principle for odd solutions of a nonlocal parabolic equation in [JLX], and in their other paper [JX] (Theorem 1.8), they obtained classiﬁcations of solutions for (3”
Blow-up analysis of a nonlocal Liouville-type equation (2015, background): “We mention that a nonlocal version of the Nirenberg problem in dimension n≥2 has recently been studied in [Jin et al. 2014; 2015a].”
Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations (2017, background): “s Theorem 5.1 below. The latter is a basic H¨older continuity theorem of De Giorgi-Nash-Moser type for the extension problem. For the elliptic counterpart of such result one should see [CS], but also [JLX], [TX]. Henceforth, for a given domain Ω ⊂ Rn+1, and 0 <α≤ 1, we denote by Hα(Ω) the non-iso”
Monotonicity and nonexistence results for some fractional elliptic problems in the half-space (2013, result): “2s , then the problem (1.5) ( (−∆)su= uq, u≥ 0 in RN +, u= 0 in RN\ RN +. only admits the trivial solution u≡ 0. Our results complement the following recent Liouville type result of Jin, Li and Xiong [27] for the corresponding full space problem (1.6) (−∆)su= uq, u>0 in RN. Theorem 1.3 (see [27])”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “Namely, the power-like nonlinearity up is inherited by Riemannian geometry and the critical exponent p = n+2s n−2s comes from conformal invariance in the classical case (see also the recent fractional contributions in [34, 35] in which the same term is taken into account).”
Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities (2019, unknown): “6 in [25]), we have sup ∂′′B+ Vj ≤ C( inf ∂′′B+ Vj + 1), where ∂′′B+ = {(x, t) : t > 0, |x|2 + t2 = 1}.”
Delaunay-type singular solutions for the fractional Yamabe problem (2015, background): “3) have been completely classified ([19, 10]) and, in particular, they must be the standard “bubbles” w(x) = c (”
Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities (2013, background): “say that U is a nonnegative solution of (2) if U is in the weighted Sobolev space W1,2(t1−2σ,B+ 2 \ B + ε ) for all ε > 0, U ≥ 0, and it satisﬁes ( 2) in the sense of distribution away from 0 (see [13] for more details on this deﬁnition). Then it follows from th e regularity result in [13] that U”