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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/s00039-014-0298-z. Google Scholar

[2]

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three-dimensional sphere,, J. Funct. Anal., 95 (1991), 106. doi: 10.1016/0022-1236(91)90026-2. Google Scholar

[3]

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

[4]

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

[5]

P. Billingsley, Convergence of Probability Measures,, J. Wiley and Sons, (1968). Google Scholar

[6]

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

[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., (1999). Google Scholar

[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. doi: 10.3934/dcds.2001.7.275. Google Scholar

[9]

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

[10]

S. Y. A. Chang and P. Yang, The Q-curvature equation in conformal geometry, Géométrie différentielle,, physique mathématique, 322 (2008), 23. Google Scholar

[11]

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

[12]

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

[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. doi: 10.1016/j.crma.2010.12.016. Google Scholar

[14]

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

[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. doi: 10.1215/S0012-7094-93-07117-7. Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[19]

J. Glimm and A. Jaffe, Quantum Physics,, $2^{nd}$ ed., (1987). Google Scholar

[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. doi: 10.1112/jlms/s2-46.3.557. Google Scholar

[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. doi: 10.1007/s00526-014-0718-9. Google Scholar

[22]

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

[23]

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

[24]

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

[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). Google Scholar

[26]

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

[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. doi: 10.4171/RLM/577. Google Scholar

[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. doi: 10.1007/s00039-011-0134-7. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

Y. Wang, Curvature and Statistics,, Ph.D. Dissertation, (2013). Google Scholar

[35]

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

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. doi: 10.1007/s00039-014-0298-z. Google Scholar

[2]

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three-dimensional sphere,, J. Funct. Anal., 95 (1991), 106. doi: 10.1016/0022-1236(91)90026-2. Google Scholar

[3]

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

[4]

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

[5]

P. Billingsley, Convergence of Probability Measures,, J. Wiley and Sons, (1968). Google Scholar

[6]

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

[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., (1999). Google Scholar

[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. doi: 10.3934/dcds.2001.7.275. Google Scholar

[9]

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

[10]

S. Y. A. Chang and P. Yang, The Q-curvature equation in conformal geometry, Géométrie différentielle,, physique mathématique, 322 (2008), 23. Google Scholar

[11]

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

[12]

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

[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. doi: 10.1016/j.crma.2010.12.016. Google Scholar

[14]

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

[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. doi: 10.1215/S0012-7094-93-07117-7. Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[19]

J. Glimm and A. Jaffe, Quantum Physics,, $2^{nd}$ ed., (1987). Google Scholar

[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. doi: 10.1112/jlms/s2-46.3.557. Google Scholar

[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. doi: 10.1007/s00526-014-0718-9. Google Scholar

[22]

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

[23]

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

[24]

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

[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). Google Scholar

[26]

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

[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. doi: 10.4171/RLM/577. Google Scholar

[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. doi: 10.1007/s00039-011-0134-7. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

Y. Wang, Curvature and Statistics,, Ph.D. Dissertation, (2013). Google Scholar

[35]

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

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