Advanced Search
Article Contents
Article Contents

Concentration of solutions for the fractional Nirenberg problem

Abstract Related Papers Cited by
  • The aim of this paper is to show the existence of infinitely many concentration solutions for the fractional Nirenberg problem under the condition that $Q_s$ curvature has a sequence of strictly local maximum points moving to infinity.
    Mathematics Subject Classification: Primary: 35J60, 53C21; Secondary: 35R09.


    \begin{equation} \\ \end{equation}
  • [1]

    W. Abdelhedi and H. Chtioui, On a Nirenberg-type problem involving the square root of the Laplacian, J. Funct. Anal., 265 (2013), 2937-2955.doi: 10.1016/j.jfa.2013.08.005.


    A. Bahri, Critical Points at Infnity in Some Variational Problems, Research Notes in Mathematics, Vol.182, Longman-Pitman, 1989.doi: 0-582-02164-2.


    A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology on the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.doi: 10.1002/cpa.3160410302.


    A. Bahri and J. M. Coron, The scalar crvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172.doi: 10.1016/0022-1236(91)90026-2.


    X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.doi: 10.1016/j.anihpc.2013.02.001.


    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.doi: 10.1080/03605300600987306.


    D. M. Cao, E. S. Noussair and S. Yan, On the scalar curvature equation $-\Delta u=(1+\varepsilon K)u^{(N+2)/(N-2)}$ in $\mathbbR^N$, Calc. Var. Partial Differential Equations, 15 (2002), 403-419.doi: 10.1007/s00526-002-0137-1.


    D. M. Cao and S. J. Peng, Solutions for the Prescribing Mean Curvature equation, Acta Math. Appl. Sinica, English Series, 24 (2008), 497-510.doi: 10.1007/s10255-008-8051-8.


    D. M. Cao and S. J. Peng, Concentration of solutions for the Yamabe problem on half-spaces, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 73-99.doi: 10.1017/S0308210511000291.


    D. M. Cao, S. J. Peng and S. Yan, On the Webster scalar curvature problem on the CR sphere with a cylindrical- type symmetry, J. Geom. Anal., 23 (2013), 1674-1702.doi: 10.1007/s12220-012-9301-9.


    S. Y. Alice Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.doi: 10.1016/j.aim.2010.07.016.


    W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.doi: 10.1002/cpa.20116.


    G. Chen and Y. Zheng, A perturbation result for the $Q_\gamma$ curvature problem on $S^n$, Nonlinear Anal., 97 (2014), 4-14.doi: 10.1016/j.na.2013.11.010.


    G. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem, Nonlinear Anal., 122 (2015), 100-124.


    J. Dávila, M. del Pino and Y. Sire, Nondegeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 141 (2013), 3865-3870.doi: 10.1090/S0002-9939-2013-12177-5.


    Z. Djadli, E. Hebey and M. Ledoux, Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169.doi: 10.1215/S0012-7094-00-10416-4.


    M. del Mar González, R. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863.doi: 10.1007/s12220-011-9217-9.


    M. del Mar González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535-1576.doi: 10.2140/apde.2013.6.1535.


    E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.doi: 10.1016/j.bulsci.2011.12.004.


    C. R. Graham, R. Jenne, L. J. Mason and G. A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. Lond. Math. Soc., 46 (1992), 557-565.doi: 10.1112/jlms/s2-46.3.557.


    C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.doi: 10.1007/s00222-002-0268-1.


    T. Jin, Y. Li and J. Xiong, The Nirenberg problem and its generalizations: A unified approach, preprint, arXiv:math/1411.5743.


    T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.doi: 10.4171/JEMS/456.


    T. Jin, Y. Li and J. Xiong, On a fractional nirenberg problem, part II: Existence of solutions, Int. Math. Res. Not., 2015 (2015), 1555-1589.


    Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.


    Y. Li, P. Mastrolia and D. D. Monticelli, On conformally invariant equations on $R^n$-II. Exponential invariance, Nonlinear Anal., 75 (2012), 5194-5211.doi: 10.1016/j.na.2012.04.036.


    Y. Li, P. Mastrolia and D. D. Monticelli, On conformally invariant equations on $R^n$, Nonlinear Anal., 95 (2014), 339-361.doi: 10.1016/j.na.2013.09.016.


    E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.doi: 10.2307/2007032.


    Z. Liu, Concentration phenomena for the Paneitz curvature equation in $\mathbbR^N$, Adv. Nonlinear Stud., 13 (2013), 837-851.


    Z. Liu, Arbitrary many peak solutions for a bi-harmonic equation with nearly critical growth, J. Math. Anal. Appl., 398 (2013), 671-691.


    S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 036. 3.doi: 10.3842/SIGMA.2008.036.


    S. J. Peng and J. Zhou, Concentration of solutions for a Paneitz type problem, Discrete Continuous Dynamic Systems, 26 (2010), 1055-1072.doi: 10.3934/dcds.2010.26.1055.


    L. J. Peterson, Conformally covariant pseudo-differential operators, Differential Geom. Appl., 13 (2000), 197-211.doi: 10.1016/S0926-2245(00)00023-1.


    O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.doi: 10.1016/0022-1236(90)90002-3.


    S. Yan, Concentration of solutions for the scalar curvature equation on $R^N$, J. Differential Equations, 163 (2000), 239-264.doi: 10.1006/jdeq.1999.3718.

  • 加载中

Article Metrics

HTML views() PDF downloads(192) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint