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

Chungen Liu; Yafang Wang

doi:10.3934/cpaa.2018043

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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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