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May  2018, 17(3): 849-885. doi: 10.3934/cpaa.2018043

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

Chungen Liu 1,, and  Yafang Wang 2,

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2. 

School of Mathematical Sciences, Nankai University, Tianjin 300071, China

* Corresponding author

Received  July 2017 Revised  July 2017 Published  January 2018

Fund Project: The first author is supported by NSF of China (11471170).

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.

Keywords: CR manifolds, CR fractional sub-Laplacian, Yamabe problem, critical points at infinity, fractional Q-curvature.
Mathematics Subject Classification: Primary: 35H20, 35S05; Secondary: 53C55.
Citation: Chungen Liu, Yafang Wang. Existence results for the fractional Q-curvature problem on three dimensional CR sphere. Communications on Pure & Applied Analysis, 2018, 17 (3) : 849-885. doi: 10.3934/cpaa.2018043
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, Vol. 65. Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$, vol. 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006. Google Scholar

[3]

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. Google Scholar

[4]

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 Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.   Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

T. Bieske, On ∞-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727-761.  doi: 10.1081/PDE-120002872.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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. Google Scholar

[16]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.   Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.  Google Scholar

[20]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear schrödinger equations, Communications on Pure and Applied Analysis, 13 (2014), 2359-2376.   Google Scholar

[21]

G. Chen and Y. Zheng, A perturbation result for the curvature problem on $S^n$, Nonlinear Analysis, 97 (2014), 4-14.   Google Scholar

[22]

W. X. Chen, Scalar curvatures on $S^n$, Math. Ann., 283 (1989), 353-365.  doi: 10.1007/BF01442733.  Google Scholar

[23]

W. X. Chen and W. Y. Ding, Scalar curvatures on $S^2$, Trans. Amer. Math. Soc., 303 (1987), 365-382.  doi: 10.2307/2000798.  Google Scholar

[24]

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.  Google Scholar

[25]

S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, vol. 246 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 2006. Google Scholar

[26]

J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.  doi: 10.1007/BF01389071.  Google Scholar

[27]

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.   Google Scholar

[28]

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.   Google Scholar

[29]

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.  Google Scholar

[30]

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.  Google Scholar

[31]

N. Gamara, The CR Yamabe conjecture-the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137.  doi: 10.1007/PL00011303.  Google Scholar

[32]

N. Gamara, The prescribed scalar curvature on a 3-dimensional CR manifold, Adv. Nonlinear Stud., 2 (2002), 193-235.   Google Scholar

[33]

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.  Google Scholar

[34]

Z.-C. Han, Prescribing Gaussian curvature on $S^2$, Duke Math. J., 61 (1990), 679-703.  doi: 10.1215/S0012-7094-90-06125-3.  Google Scholar

[35]

E. Hebey, Changements de métriques conformes sur la sphére. Le probléme de Nirenberg, Bull. Sci. Math., 114 (1990), 215-242.   Google Scholar

[36]

D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.   Google Scholar

[37]

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.  Google Scholar

[38]

D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343.   Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

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.  Google Scholar

[42]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. Google Scholar

[43]

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.  Google Scholar

[44]

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.  Google Scholar

[45]

J. Moser, On a nonlinear problem in differential geometry, 273-280. Google Scholar

[46]

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.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, Vol. 65. Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$, vol. 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006. Google Scholar

[3]

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. Google Scholar

[4]

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 Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.   Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

T. Bieske, On ∞-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727-761.  doi: 10.1081/PDE-120002872.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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. Google Scholar

[16]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.   Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.  Google Scholar

[20]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear schrödinger equations, Communications on Pure and Applied Analysis, 13 (2014), 2359-2376.   Google Scholar

[21]

G. Chen and Y. Zheng, A perturbation result for the curvature problem on $S^n$, Nonlinear Analysis, 97 (2014), 4-14.   Google Scholar

[22]

W. X. Chen, Scalar curvatures on $S^n$, Math. Ann., 283 (1989), 353-365.  doi: 10.1007/BF01442733.  Google Scholar

[23]

W. X. Chen and W. Y. Ding, Scalar curvatures on $S^2$, Trans. Amer. Math. Soc., 303 (1987), 365-382.  doi: 10.2307/2000798.  Google Scholar

[24]

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.  Google Scholar

[25]

S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, vol. 246 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 2006. Google Scholar

[26]

J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.  doi: 10.1007/BF01389071.  Google Scholar

[27]

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.   Google Scholar

[28]

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.   Google Scholar

[29]

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.  Google Scholar

[30]

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.  Google Scholar

[31]

N. Gamara, The CR Yamabe conjecture-the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137.  doi: 10.1007/PL00011303.  Google Scholar

[32]

N. Gamara, The prescribed scalar curvature on a 3-dimensional CR manifold, Adv. Nonlinear Stud., 2 (2002), 193-235.   Google Scholar

[33]

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.  Google Scholar

[34]

Z.-C. Han, Prescribing Gaussian curvature on $S^2$, Duke Math. J., 61 (1990), 679-703.  doi: 10.1215/S0012-7094-90-06125-3.  Google Scholar

[35]

E. Hebey, Changements de métriques conformes sur la sphére. Le probléme de Nirenberg, Bull. Sci. Math., 114 (1990), 215-242.   Google Scholar

[36]

D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.   Google Scholar

[37]

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.  Google Scholar

[38]

D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343.   Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

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.  Google Scholar

[42]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. Google Scholar

[43]

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.  Google Scholar

[44]

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.  Google Scholar

[45]

J. Moser, On a nonlinear problem in differential geometry, 273-280. Google Scholar

[46]

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.  Google Scholar

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