American Institute of Mathematical Sciences

Advanced
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, 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.  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), 6133-6162. 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), 39-71. 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), 2052-2093. 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), 23-53. 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), 403-419. 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), 27-69. 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), 115-178.  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), 67-171.  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), 113-145. 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), 521-573. 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), 809-836. 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), 1111-1171. 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), 541-597. 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), 895-924. 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), 349-374. 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), 483-514. 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), 1-25. 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), 21-41. 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), 975-983. 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), 3048-3081. 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), 239-264. 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), 47-79. 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), 106-172. 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), 6133-6162. 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), 39-71. 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), 2052-2093. 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), 23-53. 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), 403-419. 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), 27-69. 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), 115-178.  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), 67-171.  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), 113-145. 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), 521-573. 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), 809-836. 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), 1111-1171. 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), 541-597. 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), 895-924. 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), 349-374. 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), 483-514. 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), 1-25. 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), 21-41. 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), 975-983. 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), 3048-3081. 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), 239-264. 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), 47-79. doi: 10.1016/j.jfa.2015.04.012.  Google Scholar

[1]

Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461
[2]

Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239
[3]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[4]

Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603
[5]

Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321
[6]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283
[7]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[8]

Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
[9]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[10]

Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267
[11]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[12]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[13]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
[14]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[15]

Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039
[16]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058
[17]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[18]

Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
[19]

Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
[20]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (166)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy