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December  2016, 36(12): 6873-6898. doi: 10.3934/dcds.2016099

Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator

Yuxia Guo 1, and  Jianjun Nie 2,

1. 

Department of Mathematics, Tsinghua University, Beijing, 100084
2. 

Department of Mathematics, Tsinghua University, Beijing 100084, China

Received  December 2015 Revised  April 2016 Published  October 2016

We consider the following problem: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=K(y)u^{p-1} \hbox { in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right.                         (P) \end{equation*} where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geq4$ and $p=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under the condition that the function $K(y)$ has a local maximum point, we prove the existence of infinitely many non-radial solutions for the problem $(P)$.
Keywords: infinitely many solutions, elliptic equation, reduction., Fractional Laplacian, critical exponent.
Mathematics Subject Classification: 35B09, 35J6.
Citation: Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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C. C. Chen and C. S. Lin, Prescribing scalar curvature on $S^N$, I. A priori estimates,, J. Differential Geom., 57 (2001), 67.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

C. C. Chen and C. S. Lin, Prescribing scalar curvature on $S^N$, I. A priori estimates,, J. Differential Geom., 57 (2001), 67.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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