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

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 106-172. 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), 6133-6162. 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), 39-71. 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), 2052-2093. 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), 23-53. 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), 403-419. 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), 27-69. 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), 115-178.
[9]	C. C. Chen and C. S. Lin, Prescribing scalar curvature on $S^N$, I. A priori estimates, J. Differential Geom., 57 (2001), 67-171.
[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), 113-145. 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), 521-573. 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), 809-836. 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), 1111-1171. 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), 541-597. 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), 895-924. 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), 349-374. doi: 10.2307/2007032.
[17]	E. S. Noussair and S. S. Yan, The scalar curvature equation on $\mathbbR^N$, Nonlinear Anal., 45 (2001), 483-514. 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), 1-25. 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), 21-41. 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), 975-983. 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), 3048-3081. 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), 239-264. 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), 47-79. doi: 10.1016/j.jfa.2015.04.012.

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References:

[1]	A. Bahri and J. Coron, The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95 (1991), 106-172. 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), 6133-6162. 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), 39-71. 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), 2052-2093. 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), 23-53. 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), 403-419. 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), 27-69. 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), 115-178.
[9]	C. C. Chen and C. S. Lin, Prescribing scalar curvature on $S^N$, I. A priori estimates, J. Differential Geom., 57 (2001), 67-171.
[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), 113-145. 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), 521-573. 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), 809-836. 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), 1111-1171. 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), 541-597. 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), 895-924. 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), 349-374. doi: 10.2307/2007032.
[17]	E. S. Noussair and S. S. Yan, The scalar curvature equation on $\mathbbR^N$, Nonlinear Anal., 45 (2001), 483-514. 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), 1-25. 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), 21-41. 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), 975-983. 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), 3048-3081. 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), 239-264. 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), 47-79. doi: 10.1016/j.jfa.2015.04.012.

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