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March  2016, 15(2): 563-576. doi: 10.3934/cpaa.2016.15.563

Concentration of solutions for the fractional Nirenberg problem

Zhongyuan Liu 1,

1. 

School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, China

Received  July 2015 Revised  October 2015 Published  January 2016

The aim of this paper is to show the existence of infinitely many concentration solutions for the fractional Nirenberg problem under the condition that $Q_s$ curvature has a sequence of strictly local maximum points moving to infinity.
Keywords: concentration of solutions, Fractional Nirenberg problem, finite dimensional reduction..
Mathematics Subject Classification: Primary: 35J60, 53C21; Secondary: 35R0.
Citation: Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure and Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563
References:
[1]

W. Abdelhedi and H. Chtioui, On a Nirenberg-type problem involving the square root of the Laplacian, J. Funct. Anal., 265 (2013), 2937-2955. doi: 10.1016/j.jfa.2013.08.005.

[2]

A. Bahri, Critical Points at Infnity in Some Variational Problems, Research Notes in Mathematics, Vol.182, Longman-Pitman, 1989. doi: 0-582-02164-2.

[3]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology on the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.

[4]

A. Bahri and J. M. Coron, The scalar crvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172. doi: 10.1016/0022-1236(91)90026-2.

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.

[6]

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.

[7]

D. M. Cao, E. S. Noussair and S. Yan, On the scalar curvature equation $-\Delta u=(1+\varepsilon K)u^{(N+2)/(N-2)}$ in $\mathbb{R}^N2$, Calc. Var. Partial Differential Equations, 15 (2002), 403-419. doi: 10.1007/s00526-002-0137-1.

[8]

D. M. Cao and S. J. Peng, Solutions for the Prescribing Mean Curvature equation, Acta Math. Appl. Sinica, English Series, 24 (2008), 497-510. doi: 10.1007/s10255-008-8051-8.

[9]

D. M. Cao and S. J. Peng, Concentration of solutions for the Yamabe problem on half-spaces, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 73-99. doi: 10.1017/S0308210511000291.

[10]

D. M. Cao, S. J. Peng and S. Yan, On the Webster scalar curvature problem on the CR sphere with a cylindrical- type symmetry, J. Geom. Anal., 23 (2013), 1674-1702. doi: 10.1007/s12220-012-9301-9.

[11]

S. Y. Alice Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[13]

G. Chen and Y. Zheng, A perturbation result for the $Q_\gamma$ curvature problem on $S^n$, Nonlinear Anal., 97 (2014), 4-14. doi: 10.1016/j.na.2013.11.010.

[14]

G. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem, Nonlinear Anal., 122 (2015), 100-124.

[15]

J. Dávila, M. del Pino and Y. Sire, Nondegeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 141 (2013), 3865-3870. doi: 10.1090/S0002-9939-2013-12177-5.

[16]

Z. Djadli, E. Hebey and M. Ledoux, Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169. doi: 10.1215/S0012-7094-00-10416-4.

[17]

M. del Mar González, R. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863. doi: 10.1007/s12220-011-9217-9.

[18]

M. del Mar González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535-1576. doi: 10.2140/apde.2013.6.1535.

[19]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[20]

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

[21]

C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118. doi: 10.1007/s00222-002-0268-1.

[22]

T. Jin, Y. Li and J. Xiong, The Nirenberg problem and its generalizations: A unified approach, preprint, arXiv:math/1411.5743.

[23]

T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.

[24]

T. Jin, Y. Li and J. Xiong, On a fractional nirenberg problem, part II: Existence of solutions, Int. Math. Res. Not., 2015 (2015), 1555-1589.

[25]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.

[26]

Y. Li, P. Mastrolia and D. D. Monticelli, On conformally invariant equations on $R^n$-II. Exponential invariance, Nonlinear Anal., 75 (2012), 5194-5211. doi: 10.1016/j.na.2012.04.036.

[27]

Y. Li, P. Mastrolia and D. D. Monticelli, On conformally invariant equations on $R^n$, Nonlinear Anal., 95 (2014), 339-361. doi: 10.1016/j.na.2013.09.016.

[28]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.

[29]

Z. Liu, Concentration phenomena for the Paneitz curvature equation in $\mathbb{R}^N2$, Adv. Nonlinear Stud., 13 (2013), 837-851.

[30]

Z. Liu, Arbitrary many peak solutions for a bi-harmonic equation with nearly critical growth, J. Math. Anal. Appl., 398 (2013), 671-691.

[31]

S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 036. 3. doi: 10.3842/SIGMA.2008.036.

[32]

S. J. Peng and J. Zhou, Concentration of solutions for a Paneitz type problem, Discrete Continuous Dynamic Systems, 26 (2010), 1055-1072. doi: 10.3934/dcds.2010.26.1055.

[33]

L. J. Peterson, Conformally covariant pseudo-differential operators, Differential Geom. Appl., 13 (2000), 197-211. doi: 10.1016/S0926-2245(00)00023-1.

[34]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.

[35]

S. Yan, Concentration of solutions for the scalar curvature equation on $R^N$, J. Differential Equations, 163 (2000), 239-264. doi: 10.1006/jdeq.1999.3718.

show all references

References:
[1]

W. Abdelhedi and H. Chtioui, On a Nirenberg-type problem involving the square root of the Laplacian, J. Funct. Anal., 265 (2013), 2937-2955. doi: 10.1016/j.jfa.2013.08.005.

[2]

A. Bahri, Critical Points at Infnity in Some Variational Problems, Research Notes in Mathematics, Vol.182, Longman-Pitman, 1989. doi: 0-582-02164-2.

[3]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology on the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.

[4]

A. Bahri and J. M. Coron, The scalar crvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172. doi: 10.1016/0022-1236(91)90026-2.

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.

[6]

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.

[7]

D. M. Cao, E. S. Noussair and S. Yan, On the scalar curvature equation $-\Delta u=(1+\varepsilon K)u^{(N+2)/(N-2)}$ in $\mathbb{R}^N2$, Calc. Var. Partial Differential Equations, 15 (2002), 403-419. doi: 10.1007/s00526-002-0137-1.

[8]

D. M. Cao and S. J. Peng, Solutions for the Prescribing Mean Curvature equation, Acta Math. Appl. Sinica, English Series, 24 (2008), 497-510. doi: 10.1007/s10255-008-8051-8.

[9]

D. M. Cao and S. J. Peng, Concentration of solutions for the Yamabe problem on half-spaces, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 73-99. doi: 10.1017/S0308210511000291.

[10]

D. M. Cao, S. J. Peng and S. Yan, On the Webster scalar curvature problem on the CR sphere with a cylindrical- type symmetry, J. Geom. Anal., 23 (2013), 1674-1702. doi: 10.1007/s12220-012-9301-9.

[11]

S. Y. Alice Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[13]

G. Chen and Y. Zheng, A perturbation result for the $Q_\gamma$ curvature problem on $S^n$, Nonlinear Anal., 97 (2014), 4-14. doi: 10.1016/j.na.2013.11.010.

[14]

G. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem, Nonlinear Anal., 122 (2015), 100-124.

[15]

J. Dávila, M. del Pino and Y. Sire, Nondegeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 141 (2013), 3865-3870. doi: 10.1090/S0002-9939-2013-12177-5.

[16]

Z. Djadli, E. Hebey and M. Ledoux, Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169. doi: 10.1215/S0012-7094-00-10416-4.

[17]

M. del Mar González, R. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863. doi: 10.1007/s12220-011-9217-9.

[18]

M. del Mar González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535-1576. doi: 10.2140/apde.2013.6.1535.

[19]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[20]

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

[21]

C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118. doi: 10.1007/s00222-002-0268-1.

[22]

T. Jin, Y. Li and J. Xiong, The Nirenberg problem and its generalizations: A unified approach, preprint, arXiv:math/1411.5743.

[23]

T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.

[24]

T. Jin, Y. Li and J. Xiong, On a fractional nirenberg problem, part II: Existence of solutions, Int. Math. Res. Not., 2015 (2015), 1555-1589.

[25]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.

[26]

Y. Li, P. Mastrolia and D. D. Monticelli, On conformally invariant equations on $R^n$-II. Exponential invariance, Nonlinear Anal., 75 (2012), 5194-5211. doi: 10.1016/j.na.2012.04.036.

[27]

Y. Li, P. Mastrolia and D. D. Monticelli, On conformally invariant equations on $R^n$, Nonlinear Anal., 95 (2014), 339-361. doi: 10.1016/j.na.2013.09.016.

[28]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.

[29]

Z. Liu, Concentration phenomena for the Paneitz curvature equation in $\mathbb{R}^N2$, Adv. Nonlinear Stud., 13 (2013), 837-851.

[30]

Z. Liu, Arbitrary many peak solutions for a bi-harmonic equation with nearly critical growth, J. Math. Anal. Appl., 398 (2013), 671-691.

[31]

S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 036. 3. doi: 10.3842/SIGMA.2008.036.

[32]

S. J. Peng and J. Zhou, Concentration of solutions for a Paneitz type problem, Discrete Continuous Dynamic Systems, 26 (2010), 1055-1072. doi: 10.3934/dcds.2010.26.1055.

[33]

L. J. Peterson, Conformally covariant pseudo-differential operators, Differential Geom. Appl., 13 (2000), 197-211. doi: 10.1016/S0926-2245(00)00023-1.

[34]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.

[35]

S. Yan, Concentration of solutions for the scalar curvature equation on $R^N$, J. Differential Equations, 163 (2000), 239-264. doi: 10.1006/jdeq.1999.3718.

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