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

Concentration of solutions for the fractional Nirenberg problem

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.
Citation: Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure & 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,, \emph{J. Funct. Anal.}, 265 (2013), 2937. doi: 10.1016/j.jfa.2013.08.005. Google Scholar

[2]

A. Bahri, Critical Points at Infnity in Some Variational Problems,, Research Notes in Mathematics, (1989). doi: 0-582-02164-2. Google Scholar

[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,, \emph{Comm. Pure Appl. Math.}, 41 (1988), 253. doi: 10.1002/cpa.3160410302. Google Scholar

[4]

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

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[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 $\mathbbR^N$,, \emph{Calc. Var. Partial Differential Equations}, 15 (2002), 403. doi: 10.1007/s00526-002-0137-1. Google Scholar

[8]

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

[9]

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

[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,, \emph{J. Geom. Anal.}, 23 (2013), 1674. doi: 10.1007/s12220-012-9301-9. Google Scholar

[11]

S. Y. Alice Chang and M. del Mar González, Fractional Laplacian in conformal geometry,, \emph{Adv. Math.}, 226 (2011), 1410. doi: 10.1016/j.aim.2010.07.016. Google Scholar

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[13]

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

[14]

G. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem,, \emph{Nonlinear Anal.}, 122 (2015), 100. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

M. del Mar González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems,, \emph{Anal. PDE}, 6 (2013), 1535. doi: 10.2140/apde.2013.6.1535. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

T. Jin, Y. Li and J. Xiong, The Nirenberg problem and its generalizations: A unified approach,, preprint, (). Google Scholar

[23]

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

[24]

T. Jin, Y. Li and J. Xiong, On a fractional nirenberg problem, part II: Existence of solutions,, \emph{Int. Math. Res. Not.}, 2015 (2015), 1555. Google Scholar

[25]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres,, \emph{J. Eur. Math. Soc.}, 6 (2004), 153. Google Scholar

[26]

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

[27]

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

[28]

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

[29]

Z. Liu, Concentration phenomena for the Paneitz curvature equation in $\mathbbR^N$,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 837. Google Scholar

[30]

Z. Liu, Arbitrary many peak solutions for a bi-harmonic equation with nearly critical growth,, \emph{J. Math. Anal. Appl.}, 398 (2013), 671. Google Scholar

[31]

S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary),, \emph{SIGMA Symmetry Integrability Geom. Methods Appl.}, 4 (2008). doi: 10.3842/SIGMA.2008.036. Google Scholar

[32]

S. J. Peng and J. Zhou, Concentration of solutions for a Paneitz type problem,, \emph{Discrete Continuous Dynamic Systems}, 26 (2010), 1055. doi: 10.3934/dcds.2010.26.1055. Google Scholar

[33]

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

[34]

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

[35]

S. Yan, Concentration of solutions for the scalar curvature equation on $R^N$,, \emph{J. Differential Equations}, 163 (2000), 239. doi: 10.1006/jdeq.1999.3718. Google Scholar

show all references

References:
[1]

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

[2]

A. Bahri, Critical Points at Infnity in Some Variational Problems,, Research Notes in Mathematics, (1989). doi: 0-582-02164-2. Google Scholar

[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,, \emph{Comm. Pure Appl. Math.}, 41 (1988), 253. doi: 10.1002/cpa.3160410302. Google Scholar

[4]

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

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[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 $\mathbbR^N$,, \emph{Calc. Var. Partial Differential Equations}, 15 (2002), 403. doi: 10.1007/s00526-002-0137-1. Google Scholar

[8]

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

[9]

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

[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,, \emph{J. Geom. Anal.}, 23 (2013), 1674. doi: 10.1007/s12220-012-9301-9. Google Scholar

[11]

S. Y. Alice Chang and M. del Mar González, Fractional Laplacian in conformal geometry,, \emph{Adv. Math.}, 226 (2011), 1410. doi: 10.1016/j.aim.2010.07.016. Google Scholar

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[13]

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

[14]

G. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem,, \emph{Nonlinear Anal.}, 122 (2015), 100. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

M. del Mar González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems,, \emph{Anal. PDE}, 6 (2013), 1535. doi: 10.2140/apde.2013.6.1535. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

T. Jin, Y. Li and J. Xiong, The Nirenberg problem and its generalizations: A unified approach,, preprint, (). Google Scholar

[23]

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

[24]

T. Jin, Y. Li and J. Xiong, On a fractional nirenberg problem, part II: Existence of solutions,, \emph{Int. Math. Res. Not.}, 2015 (2015), 1555. Google Scholar

[25]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres,, \emph{J. Eur. Math. Soc.}, 6 (2004), 153. Google Scholar

[26]

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

[27]

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

[28]

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

[29]

Z. Liu, Concentration phenomena for the Paneitz curvature equation in $\mathbbR^N$,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 837. Google Scholar

[30]

Z. Liu, Arbitrary many peak solutions for a bi-harmonic equation with nearly critical growth,, \emph{J. Math. Anal. Appl.}, 398 (2013), 671. Google Scholar

[31]

S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary),, \emph{SIGMA Symmetry Integrability Geom. Methods Appl.}, 4 (2008). doi: 10.3842/SIGMA.2008.036. Google Scholar

[32]

S. J. Peng and J. Zhou, Concentration of solutions for a Paneitz type problem,, \emph{Discrete Continuous Dynamic Systems}, 26 (2010), 1055. doi: 10.3934/dcds.2010.26.1055. Google Scholar

[33]

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

[34]

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

[35]

S. Yan, Concentration of solutions for the scalar curvature equation on $R^N$,, \emph{J. Differential Equations}, 163 (2000), 239. doi: 10.1006/jdeq.1999.3718. Google Scholar

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