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

Chungen Liu; Yafang Wang

doi:10.3934/cpaa.2018043

Article Contents

2018, Volume 17, Issue 3: 849-885. Doi: 10.3934/cpaa.2018043

This issue Previous Article The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium Next Article Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

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
* Corresponding author

Received: June 30, 2017

Revised: June 30, 2017

Published: May 2018

The first author is supported by NSF of China (11471170)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	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.