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The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium
Existence results for the fractional Q-curvature problem on three dimensional CR sphere
1. | School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China |
2. | School of Mathematical Sciences, Nankai University, Tianjin 300071, China |
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.
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. |
[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. |
[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. |
[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 |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[11] |
T. Bieske,
On ∞-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727-761.
doi: 10.1081/PDE-120002872. |
[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. |
[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. |
[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. |
[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. |
[16] |
S.-Y. A. Chang and P. C. Yang,
Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.
|
[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. |
[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. |
[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. |
[20] |
G. Chen and Y. Zheng,
Concentration phenomenon for fractional nonlinear schrödinger equations, Communications on Pure and Applied Analysis, 13 (2014), 2359-2376.
|
[21] |
G. Chen and Y. Zheng,
A perturbation result for the curvature problem on $S^n$, Nonlinear Analysis, 97 (2014), 4-14.
|
[22] |
W. X. Chen,
Scalar curvatures on $S^n$, Math. Ann., 283 (1989), 353-365.
doi: 10.1007/BF01442733. |
[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. |
[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. |
[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. |
[26] |
J. F. Escobar and R. M. Schoen,
Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.
doi: 10.1007/BF01389071. |
[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.
|
[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.
|
[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. |
[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. |
[31] |
N. Gamara,
The CR Yamabe conjecture-the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137.
doi: 10.1007/PL00011303. |
[32] |
N. Gamara,
The prescribed scalar curvature on a 3-dimensional CR manifold, Adv. Nonlinear Stud., 2 (2002), 193-235.
|
[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. |
[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. |
[35] |
E. Hebey,
Changements de métriques conformes sur la sphére. Le probléme de Nirenberg, Bull. Sci. Math., 114 (1990), 215-242.
|
[36] |
D. Jerison and J. M. Lee,
The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.
|
[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. |
[38] |
D. Jerison and J. M. Lee,
Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343.
|
[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. |
[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. |
[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. |
[42] |
E. H. Lieb and M. Loss,
Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. |
[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. |
[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. |
[45] |
J. Moser, On a nonlinear problem in differential geometry, 273-280. |
[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. |
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. |
[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. |
[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. |
[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 |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[11] |
T. Bieske,
On ∞-harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727-761.
doi: 10.1081/PDE-120002872. |
[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. |
[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. |
[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. |
[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. |
[16] |
S.-Y. A. Chang and P. C. Yang,
Conformal deformation of metrics on $S^2$, J. Differential Geom., 27 (1988), 259-296.
|
[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. |
[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. |
[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. |
[20] |
G. Chen and Y. Zheng,
Concentration phenomenon for fractional nonlinear schrödinger equations, Communications on Pure and Applied Analysis, 13 (2014), 2359-2376.
|
[21] |
G. Chen and Y. Zheng,
A perturbation result for the curvature problem on $S^n$, Nonlinear Analysis, 97 (2014), 4-14.
|
[22] |
W. X. Chen,
Scalar curvatures on $S^n$, Math. Ann., 283 (1989), 353-365.
doi: 10.1007/BF01442733. |
[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. |
[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. |
[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. |
[26] |
J. F. Escobar and R. M. Schoen,
Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.
doi: 10.1007/BF01389071. |
[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.
|
[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.
|
[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. |
[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. |
[31] |
N. Gamara,
The CR Yamabe conjecture-the case $n = 1$, J. Eur. Math. Soc. (JEMS), 3 (2001), 105-137.
doi: 10.1007/PL00011303. |
[32] |
N. Gamara,
The prescribed scalar curvature on a 3-dimensional CR manifold, Adv. Nonlinear Stud., 2 (2002), 193-235.
|
[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. |
[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. |
[35] |
E. Hebey,
Changements de métriques conformes sur la sphére. Le probléme de Nirenberg, Bull. Sci. Math., 114 (1990), 215-242.
|
[36] |
D. Jerison and J. M. Lee,
The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167-197.
|
[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. |
[38] |
D. Jerison and J. M. Lee,
Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343.
|
[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. |
[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. |
[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. |
[42] |
E. H. Lieb and M. Loss,
Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. |
[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. |
[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. |
[45] |
J. Moser, On a nonlinear problem in differential geometry, 273-280. |
[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. |
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