November  2016, 36(11): 6307-6330. doi: 10.3934/dcds.2016074

Prescribing the Q-curvature on the sphere with conical singularities

1. 

Department of mathematics and natural sciences, American University of Ras Al Khaimah, PO Box 10021, Ras Al Khaimah, United Arab Emirates

Received  December 2015 Revised  July 2016 Published  August 2016

In this paper we investigate the problem of prescribing the $Q$-curvature, on the sphere of any dimension with prescribed conical singularities. We also give the asymptotic behaviour of the solutions that we find and we prove their uniqueness in the negative curvature case. We focus mainly on the odd dimensional case, more specifically the three dimensional sphere.
Citation: Ali Maalaoui. Prescribing the Q-curvature on the sphere with conical singularities. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6307-6330. doi: 10.3934/dcds.2016074
References:
[1]

K. Akutagawa, G. Carron and R. Mazzeo, The Yamabe problem on stratified spaces, Geometric and Functional Analusis, 24 (2014), 1039-1079. doi: 10.1007/s00039-014-0298-z.

[2]

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.

[3]

D. Bartolucci, F. De Marchis and A. Malchiodi, Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not., 24 (2011), 5625-5643. doi: 10.1093/imrn/rnq285.

[4]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242. doi: 10.2307/2946638.

[5]

P. Billingsley, Convergence of Probability Measures, J. Wiley and Sons, New York, 1968.

[6]

T. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal., 74 (1987) , 199-291 . doi: 10.1016/0022-1236(87)90025-5.

[7]

S. Y. A. Chang, On a fourth-order partial differential equation in conformal geometry harmonic analysis and partial differential equations, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999.

[8]

S.-Y. A Chang and W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dynam. Systems, 7 (2001), 275-281. doi: 10.3934/dcds.2001.7.275.

[9]

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

[10]

S. Y. A. Chang and P. Yang, The Q-curvature equation in conformal geometry, Géométrie différentielle, physique mathématique, mathématiques et société. II. Astérisque, 322 (2008), 23-38.

[11]

S. Chanillo and M. K.-H. Kiessling, Surfaces with prescribed Gauss curvature, Duke Math. J., 105 (2000), 309-353. doi: 10.1215/S0012-7094-00-10525-X.

[12]

A. Carlotto and A. Malchiodi, Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262 (2012), 409-450. doi: 10.1016/j.jfa.2011.09.012.

[13]

A. Carlotto and A. Malchiodi, A class of existence results for the singular Liouville equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 161-166. doi: 10.1016/j.crma.2010.12.016.

[14]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.

[15]

W. Chen and C. Li, Qualitative Properties of solutions to some non-linear elliptic equations in $\mathbbR^{2}$, Duke Math. J., 71 (1993), 427-439. doi: 10.1215/S0012-7094-93-07117-7.

[16]

Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. of Math., 168 (2008), 813-858. doi: 10.4007/annals.2008.168.813.

[17]

J. Dolbeault, M. J. Esteban and G. Tarantello, The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci., VII (2008), 313-341.

[18]

R.S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8533-2.

[19]

J. Glimm and A. Jaffe, Quantum Physics, $2^{nd}$ ed., Springer Verlag, New York, 1987.

[20]

C. Graham, R. Jenne, L. Mason and G. Sparling, Conformally invariant powers of the Laplacian, I: Existence, J. London Math. Soc., 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557.

[21]

T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local $Q$-curvature equation in dimension three, Calc. Var. Partial Differential Equations, 52 (2015), 469-488. doi: 10.1007/s00526-014-0718-9.

[22]

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-42. doi: 10.2307/1971012.

[23]

M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103.

[24]

M. K.-H. Kiessling, Statistical mechanics approach to some problems in conformal geometry, Physica A, 279 (2000), 353-368. doi: 10.1016/S0378-4371(99)00515-4.

[25]

M. K.-H. Kiessling, Typicality analysis for the Newtonian N-body problem on $S^2$ in the $N\to \infty$ limit, J. Stat. Mech. Theory Exp., 01 (2011).

[26]

A. Malchiodi, Conformal metrics with constant Q-curvature, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 120, 11 pp. doi: 10.3842/SIGMA.2007.120.

[27]

A. Malchiodi, Variational methods for singular Liouville equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21 (2010), 349-358. doi: 10.4171/RLM/577.

[28]

A. Malchiodi and D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geometric and Functional Analysis, 21 (2011), 1196-1217. doi: 10.1007/s00039-011-0134-7.

[29]

L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $\mathbbR^{2m}$, Math. Z., 263 (2009), 307-329. doi: 10.1007/s00209-008-0419-1.

[30]

J. Messer and H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation, J. Stat. Phys., 29 (1982), 561-578. doi: 10.1007/BF01342187.

[31]

C. B. Ndiaye, Constant T-curvature conformal metrics on 4-manifolds with boundary, Pacific J. Math., 240 (2009), 151-184. doi: 10.2140/pjm.2009.240.151.

[32]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[33]

M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc., 324 (1991), 793-821. doi: 10.1090/S0002-9947-1991-1005085-9.

[34]

Y. Wang, Curvature and Statistics, Ph.D. Dissertation, Rutgers University, 2013.

[35]

J. Wei and X. Xu, On conformal deformations of metrics on $S^n$, J. Funct. Anal., 157 (1998), 292-325. doi: 10.1006/jfan.1998.3271.

show all references

References:
[1]

K. Akutagawa, G. Carron and R. Mazzeo, The Yamabe problem on stratified spaces, Geometric and Functional Analusis, 24 (2014), 1039-1079. doi: 10.1007/s00039-014-0298-z.

[2]

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.

[3]

D. Bartolucci, F. De Marchis and A. Malchiodi, Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not., 24 (2011), 5625-5643. doi: 10.1093/imrn/rnq285.

[4]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242. doi: 10.2307/2946638.

[5]

P. Billingsley, Convergence of Probability Measures, J. Wiley and Sons, New York, 1968.

[6]

T. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal., 74 (1987) , 199-291 . doi: 10.1016/0022-1236(87)90025-5.

[7]

S. Y. A. Chang, On a fourth-order partial differential equation in conformal geometry harmonic analysis and partial differential equations, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999.

[8]

S.-Y. A Chang and W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dynam. Systems, 7 (2001), 275-281. doi: 10.3934/dcds.2001.7.275.

[9]

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

[10]

S. Y. A. Chang and P. Yang, The Q-curvature equation in conformal geometry, Géométrie différentielle, physique mathématique, mathématiques et société. II. Astérisque, 322 (2008), 23-38.

[11]

S. Chanillo and M. K.-H. Kiessling, Surfaces with prescribed Gauss curvature, Duke Math. J., 105 (2000), 309-353. doi: 10.1215/S0012-7094-00-10525-X.

[12]

A. Carlotto and A. Malchiodi, Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262 (2012), 409-450. doi: 10.1016/j.jfa.2011.09.012.

[13]

A. Carlotto and A. Malchiodi, A class of existence results for the singular Liouville equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 161-166. doi: 10.1016/j.crma.2010.12.016.

[14]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.

[15]

W. Chen and C. Li, Qualitative Properties of solutions to some non-linear elliptic equations in $\mathbbR^{2}$, Duke Math. J., 71 (1993), 427-439. doi: 10.1215/S0012-7094-93-07117-7.

[16]

Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. of Math., 168 (2008), 813-858. doi: 10.4007/annals.2008.168.813.

[17]

J. Dolbeault, M. J. Esteban and G. Tarantello, The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci., VII (2008), 313-341.

[18]

R.S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8533-2.

[19]

J. Glimm and A. Jaffe, Quantum Physics, $2^{nd}$ ed., Springer Verlag, New York, 1987.

[20]

C. Graham, R. Jenne, L. Mason and G. Sparling, Conformally invariant powers of the Laplacian, I: Existence, J. London Math. Soc., 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557.

[21]

T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local $Q$-curvature equation in dimension three, Calc. Var. Partial Differential Equations, 52 (2015), 469-488. doi: 10.1007/s00526-014-0718-9.

[22]

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-42. doi: 10.2307/1971012.

[23]

M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103.

[24]

M. K.-H. Kiessling, Statistical mechanics approach to some problems in conformal geometry, Physica A, 279 (2000), 353-368. doi: 10.1016/S0378-4371(99)00515-4.

[25]

M. K.-H. Kiessling, Typicality analysis for the Newtonian N-body problem on $S^2$ in the $N\to \infty$ limit, J. Stat. Mech. Theory Exp., 01 (2011).

[26]

A. Malchiodi, Conformal metrics with constant Q-curvature, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 120, 11 pp. doi: 10.3842/SIGMA.2007.120.

[27]

A. Malchiodi, Variational methods for singular Liouville equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21 (2010), 349-358. doi: 10.4171/RLM/577.

[28]

A. Malchiodi and D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geometric and Functional Analysis, 21 (2011), 1196-1217. doi: 10.1007/s00039-011-0134-7.

[29]

L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $\mathbbR^{2m}$, Math. Z., 263 (2009), 307-329. doi: 10.1007/s00209-008-0419-1.

[30]

J. Messer and H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation, J. Stat. Phys., 29 (1982), 561-578. doi: 10.1007/BF01342187.

[31]

C. B. Ndiaye, Constant T-curvature conformal metrics on 4-manifolds with boundary, Pacific J. Math., 240 (2009), 151-184. doi: 10.2140/pjm.2009.240.151.

[32]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[33]

M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc., 324 (1991), 793-821. doi: 10.1090/S0002-9947-1991-1005085-9.

[34]

Y. Wang, Curvature and Statistics, Ph.D. Dissertation, Rutgers University, 2013.

[35]

J. Wei and X. Xu, On conformal deformations of metrics on $S^n$, J. Funct. Anal., 157 (1998), 292-325. doi: 10.1006/jfan.1998.3271.

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