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

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 & 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.
[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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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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