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Existence results for the fractional Q-curvature problem on three dimensional CR sphere

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    * Corresponding author 
The first author is supported by NSF of China (11471170)
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  • In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.

    Mathematics Subject Classification: Primary: 35H20, 35S05; Secondary: 53C55.


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  •   R. A. Adams, Sobolev Spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, Vol. 65.
      A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$ , vol. 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006.
      A. Bahri, Critical Points at Infinity in Some Variational Problems, vol. 182 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
      A. Bahri and H. Brezis, Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, in Topics in Geometry, vol. 20 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1996, 1-100
      A. Bahri  and  J.-M. Coron , On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988) , 253-294.  doi: 10.1002/cpa.3160410302.
      A. Bahri  and  J.-M. Coron , The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95 (1991) , 106-172.  doi: 10.1016/0022-1236(91)90026-2.
      A. Bahri  and  P. H. Rabinowitz , Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991) , 561-649. 
      A. Bahri , An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension, Duke Math. J., 81 (1996) , 323-466.  doi: 10.1215/S0012-7094-96-08116-8.
      M. Ben Ayed , Y. Chen , H. Chtioui  and  M. Hammami , On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84 (1996) , 633-677.  doi: 10.1215/S0012-7094-96-08420-3.
      G. Bianchi  and  H. Egnell , Local existence and uniqueness of positive solutions of the equation $Δ u+(1+ε\varphi(r))u^{(n+2)/(n-2)} = 0$ , in $\textbf{R}^n$ and a related equation, in Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Gregynog, 1989), vol. 7 of Progr, Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, (1992) , 111-128.  doi: 10.1007/978-1-4612-0393-3_8.
      T. Bieske , On ∞-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002) , 727-761.  doi: 10.1081/PDE-120002872.
      T. P. Branson , L. Fontana  and  C. Morpurgo , Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Ann. of Math. (2), 177 (2013) , 1-52.  doi: 10.4007/annals.2013.177.1.1.
      H. Brezis  and  J.-M. Coron , Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89 (1985) , 21-56.  doi: 10.1007/BF00281744.
      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.
      K. -c. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston Inc., Boston, MA, 1993.
      S.-Y. A. Chang  and  P. C. Yang , Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988) , 259-296. 
      S.-Y. A. Chang  and  P. C. Yang , A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991) , 27-69.  doi: 10.1215/S0012-7094-91-06402-1.
      S.-Y. A. Chang  and  M. d. M. González , Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011) , 1410-1432.  doi: 10.1016/j.aim.2010.07.016.
      S.-Y. A. Chang  and  P. C. Yang , Prescribing Gaussian curvature on $S^2$, Acta Math., 159 (1987) , 215-259.  doi: 10.1007/BF02392560.
      G. Chen  and  Y. Zheng , Concentration phenomenon for fractional nonlinear schrödinger equations, Communications on Pure and Applied Analysis, 13 (2014) , 2359-2376. 
      G. Chen  and  Y. Zheng , A perturbation result for the curvature problem on $S^n$, Nonlinear Analysis, 97 (2014) , 4-14. 
      W. X. Chen , Scalar curvatures on $S^n$, Math. Ann., 283 (1989) , 353-365.  doi: 10.1007/BF01442733.
      W. X. Chen  and  W. Y. Ding , Scalar curvatures on $S^2$, Trans. Amer. Math. Soc., 303 (1987) , 365-382.  doi: 10.2307/2000798.
      H. Chtioui , K. El Mehdi  and  N. Gamara , The Webster scalar curvature problem on the three dimensional CR manifolds, Bull. Sci. Math., 131 (2007) , 361-374.  doi: 10.1016/j.bulsci.2006.05.003.
      S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, vol. 246 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 2006.
      J. F. Escobar  and  R. M. Schoen , Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986) , 243-254.  doi: 10.1007/BF01389071.
      G. B. Folland  and  E. M. Stein , Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974) , 429-522. 
      R. L. Frank , M. d. M. González , D. D. Monticelli  and  J. Tan , An extension problem for the CR fractional Laplacian, Advances in Mathematics, 270 (2015) , 97-137. 
      R. L. Frank  and  E. H. Lieb , Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010) , 85-99.  doi: 10.1007/s00526-009-0302-x.
      R. L. Frank  and  E. H. Lieb , Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2), 176 (2012) , 349-381.  doi: 10.4007/annals.2012.176.1.6.
      N. Gamara , The CR Yamabe conjecture-the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001) , 105-137.  doi: 10.1007/PL00011303.
      N. Gamara , The prescribed scalar curvature on a 3-dimensional CR manifold, Adv. Nonlinear Stud., 2 (2002) , 193-235. 
      N. Gamara  and  R. Yacoub , CR Yamabe conjecture-the conformally flat case, Pacific J. Math., 201 (2001) , 121-175.  doi: 10.2140/pjm.2001.201.121.
      Z.-C. Han , Prescribing Gaussian curvature on $S^2$, Duke Math. J., 61 (1990) , 679-703.  doi: 10.1215/S0012-7094-90-06125-3.
      E. Hebey , Changements de métriques conformes sur la sphére. Le probléme de Nirenberg, Bull. Sci. Math., 114 (1990) , 215-242. 
      D. Jerison  and  J. M. Lee , The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987) , 167-197. 
      D. Jerison  and  J. M. Lee , Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc., 1 (1988) , 1-13.  doi: 10.2307/1990964.
      D. Jerison  and  J. M. Lee , Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989) , 303-343. 
      J. M. Lee  and  T. H. Parker , The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987) , 37-91.  doi: 10.1090/S0273-0979-1987-15514-5.
      Y. Y. Li , Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations, 120 (1995) , 319-410.  doi: 10.1006/jdeq.1995.1115.
      E. H. Lieb , Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983) , 349-374.  doi: 10.2307/2007032.
      E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001.
      A. Malchiodi  and  F. Uguzzoni , A perturbation result for the Webster scalar curvature problem on the CR sphere, J. Math. Pures Appl. (9), 81 (2002) , 983-997.  doi: 10.1016/S0021-7824(01)01249-1.
      J. J. Manfredi  and  B. Stroffolini , A version of the Hopf-Lax formula in the Heisenberg group, Comm. Partial Differential Equations, 27 (2002) , 1139-1159.  doi: 10.1081/PDE-120004897.
      J. Moser, On a nonlinear problem in differential geometry, 273-280.
      E. Salem  and  N. Gamara , The Webster scalar curvature revisited: the case of the three dimensional CR sphere, Calc. Var. Partial Differential Equations, 42 (2011) , 107-136.  doi: 10.1007/s00526-010-0382-7.
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