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Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator
1. | Department of Mathematics, Tsinghua University, Beijing, 100084 |
2. | Department of Mathematics, Tsinghua University, Beijing 100084, China |
References:
[1] |
A. Bahri and J. 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. |
[2] |
B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133.
doi: 10.1016/j.jde.2012.02.023. |
[3] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39.
doi: 10.1017/S0308210511000175. |
[4] |
X. Cabré and J. G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.
doi: 10.1016/j.aim.2010.01.025. |
[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.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
D. M. Cao, E. Noussair and S. S. Yan, On the scalar curvature equation $-\Delta u=(1+\epsilon K)u^{(N+2)/(N-2)}$ in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 15 (2002), 403.
doi: 10.1007/s00526-002-0137-1. |
[7] |
S. A. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$,, Duke Math. J., 64 (1991), 27.
doi: 10.1215/S0012-7094-91-06402-1. |
[8] |
C. C. Chen and C. S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes. II,, J. Differential Geom., 49 (1998), 115.
|
[9] |
C. C. Chen and C. S. Lin, Prescribing scalar curvature on $S^N$, I. A priori estimates,, J. Differential Geom., 57 (2001), 67.
|
[10] |
M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem,, Calc. Var. Partial Differential Equations, 16 (2003), 113.
doi: 10.1007/s005260100142. |
[11] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.
doi: 10.1016/j.bulsci.2011.12.004. |
[12] |
Y. X. Guo and B. Li, Infinitely many solutions for the prescribed curvature problem of polyharmonic operator,, Calc. Var. Partial Differential Equations, 46 (2013), 809.
doi: 10.1007/s00526-012-0504-5. |
[13] |
T. L. Jin, Y. Y. Li and J. G. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, part I: blow up analysis and compactness of solutions., J. Eur. Math. Soc., 16 (2014), 1111.
doi: 10.4171/JEMS/456. |
[14] |
Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. II. Existence and compactness,, Comm. Pure Appl. Math., 49 (1996), 541.
doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A. |
[15] |
Y. Li and W. M. Ni, On conformal scalar curvature equations in $\mathbbR^N$,, Duke Math. J., 57 (1988), 895.
doi: 10.1215/S0012-7094-88-05740-7. |
[16] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.
doi: 10.2307/2007032. |
[17] |
E. S. Noussair and S. S. Yan, The scalar curvature equation on $\mathbbR^N$,, Nonlinear Anal., 45 (2001), 483.
doi: 10.1016/S0362-546X(99)00428-9. |
[18] |
R. Schoen and D. Zhang, Prescribed scalar curvature on the $n-$ sphere,, Calc. Var. Partial Differential Equations, 4 (1996), 1.
doi: 10.1007/BF01322307. |
[19] |
J. G. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21.
doi: 10.1007/s00526-010-0378-3. |
[20] |
J. G. Tan and J. G. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms,, Discrete Contin. Dyn. Syst., 31 (2011), 975.
doi: 10.3934/dcds.2011.31.975. |
[21] |
J. C. Wei and S. S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbbS^N$,, J. Funct. Anal., 258 (2010), 3048.
doi: 10.1016/j.jfa.2009.12.008. |
[22] |
S. S. Yan, Concentration of solutions for the scalar curvature equation on $\mathbbR^N$,, J. Differential Equations, 163 (2000), 239.
doi: 10.1006/jdeq.1999.3718. |
[23] |
S. S. Yan, J. F. Yang and X. H. Yu, Equations involving fractional Laplacian operator: compactness and application,, J. Funct. Anal., 269 (2015), 47.
doi: 10.1016/j.jfa.2015.04.012. |
show all references
References:
[1] |
A. Bahri and J. 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. |
[2] |
B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133.
doi: 10.1016/j.jde.2012.02.023. |
[3] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39.
doi: 10.1017/S0308210511000175. |
[4] |
X. Cabré and J. G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.
doi: 10.1016/j.aim.2010.01.025. |
[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.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
D. M. Cao, E. Noussair and S. S. Yan, On the scalar curvature equation $-\Delta u=(1+\epsilon K)u^{(N+2)/(N-2)}$ in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 15 (2002), 403.
doi: 10.1007/s00526-002-0137-1. |
[7] |
S. A. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$,, Duke Math. J., 64 (1991), 27.
doi: 10.1215/S0012-7094-91-06402-1. |
[8] |
C. C. Chen and C. S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes. II,, J. Differential Geom., 49 (1998), 115.
|
[9] |
C. C. Chen and C. S. Lin, Prescribing scalar curvature on $S^N$, I. A priori estimates,, J. Differential Geom., 57 (2001), 67.
|
[10] |
M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem,, Calc. Var. Partial Differential Equations, 16 (2003), 113.
doi: 10.1007/s005260100142. |
[11] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.
doi: 10.1016/j.bulsci.2011.12.004. |
[12] |
Y. X. Guo and B. Li, Infinitely many solutions for the prescribed curvature problem of polyharmonic operator,, Calc. Var. Partial Differential Equations, 46 (2013), 809.
doi: 10.1007/s00526-012-0504-5. |
[13] |
T. L. Jin, Y. Y. Li and J. G. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, part I: blow up analysis and compactness of solutions., J. Eur. Math. Soc., 16 (2014), 1111.
doi: 10.4171/JEMS/456. |
[14] |
Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. II. Existence and compactness,, Comm. Pure Appl. Math., 49 (1996), 541.
doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A. |
[15] |
Y. Li and W. M. Ni, On conformal scalar curvature equations in $\mathbbR^N$,, Duke Math. J., 57 (1988), 895.
doi: 10.1215/S0012-7094-88-05740-7. |
[16] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.
doi: 10.2307/2007032. |
[17] |
E. S. Noussair and S. S. Yan, The scalar curvature equation on $\mathbbR^N$,, Nonlinear Anal., 45 (2001), 483.
doi: 10.1016/S0362-546X(99)00428-9. |
[18] |
R. Schoen and D. Zhang, Prescribed scalar curvature on the $n-$ sphere,, Calc. Var. Partial Differential Equations, 4 (1996), 1.
doi: 10.1007/BF01322307. |
[19] |
J. G. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21.
doi: 10.1007/s00526-010-0378-3. |
[20] |
J. G. Tan and J. G. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms,, Discrete Contin. Dyn. Syst., 31 (2011), 975.
doi: 10.3934/dcds.2011.31.975. |
[21] |
J. C. Wei and S. S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbbS^N$,, J. Funct. Anal., 258 (2010), 3048.
doi: 10.1016/j.jfa.2009.12.008. |
[22] |
S. S. Yan, Concentration of solutions for the scalar curvature equation on $\mathbbR^N$,, J. Differential Equations, 163 (2000), 239.
doi: 10.1006/jdeq.1999.3718. |
[23] |
S. S. Yan, J. F. Yang and X. H. Yu, Equations involving fractional Laplacian operator: compactness and application,, J. Funct. Anal., 269 (2015), 47.
doi: 10.1016/j.jfa.2015.04.012. |
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