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September  2015, 14(5): 2009-2020. doi: 10.3934/cpaa.2015.14.2009

Multiplicity of solutions for a fractional Kirchhoff type problem

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  December 2014 Revised  March 2015 Published  June 2015

In this paper, by using the (variant) Fountain Theorem, we obtain that there are infinitely many solutions for a Kirchhoff type equation that involves a nonlocal operator.
Citation: Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009
References:
[1]

C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. Google Scholar

[3]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles,, Izv. Akad. Nauk SSSR Ser., 4 (1940), 17. Google Scholar

[4]

H. Brézis, Analyse fonctionelle. Théorie et applications,, Masson, (1983). Google Scholar

[5]

C. Chen, J. Huang and L. Liu, Multiple solutions to the nonhomogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities,, Applied Mathematics Letters, 26 (2013), 754. doi: 10.1016/j.aml.2013.02.011. Google Scholar

[6]

S. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $\mathbbR^N$,, Nonlinear Anal. RWA, 14 (2013), 1477. doi: 10.1016/j.nonrwa.2012.10.010. Google Scholar

[7]

B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, Nonlinear Anal., 71 (2009), 4883. doi: 10.1016/j.na.2009.03.065. Google Scholar

[8]

F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus,, Appl. Math. Lett., 22 (2009), 819. doi: 10.1016/j.aml.2008.06.042. Google Scholar

[9]

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

[10]

A. Fiscella, Saddle point solutions for non-local elliptic operators,, preprint, (2012). Google Scholar

[11]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator,, Nonlinear Analysis, 94 (2014), 156. doi: 10.1016/j.na.2013.08.011. Google Scholar

[12]

X. He and W. Zou, Infinitely many positive solutions of Kirchhoff type problems,, Nonlinear Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar

[13]

X. He and W. Zou, Multiplicity solutions of for a class of Kirchhoff type problems,, Acta Mathematicae Applicatae Sinica, 26 (2010), 387. doi: 10.1007/s10255-010-0005-2. Google Scholar

[14]

Kirchhoff and G. Mechanik, Teubner,, Leipzig, (1883). Google Scholar

[15]

Y. Li, F. Li and J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass,, J. Math. Anal. Appl., 410 (2014), 361. doi: 10.1016/j.jmaa.2013.08.030. Google Scholar

[16]

J. L. Lions, On some quations in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial differential Equations,, Proc. Internat. Sympos., (1977), 284. Google Scholar

[17]

D. Liu and P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation,, Nonlinear Anal., 75 (2012), 5032. doi: 10.1016/j.na.2012.04.018. Google Scholar

[18]

T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar

[19]

A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition,, Nonlinear Anal., 70 (2009), 1275. doi: 10.1016/j.na.2008.02.011. Google Scholar

[20]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar

[21]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb.(N.S.), 168 (1975), 152. Google Scholar

[22]

R. Servadei, The Yamabe equation in a non-local setting,, Advances in Nonlinear Analysis, 2 (2013), 235. doi: 10.1515/anona-2013-0008. Google Scholar

[23]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity,, Contemp. Math., 595 (2013), 317. doi: 10.1090/conm/595/11809. Google Scholar

[24]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, Rev. Mat. Iberoam., 29 (2013), 1091. doi: 10.4171/RMI/750. Google Scholar

[25]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, J. Math. Anal. Appl., 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[26]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete and Continuous Dynamical Systems, 5 (2013), 2105. Google Scholar

[27]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, Commun. Pure Appl. Anal., 12 (2013), 2445. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[29]

R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent,, preprint (2012), (2012), 12. Google Scholar

[30]

G. F. Sun and K. M. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem,, Math. Commun., 19 (2014), 183. Google Scholar

[31]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations,, Nonlinear Analysis, 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061. Google Scholar

[32]

M. Willem, Minimax Theorems,, Birkhauser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[33]

Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent,, Commun. Pure Appl. Anal., 12 (2013), 2773. doi: 10.3934/cpaa.2013.12.2773. Google Scholar

[34]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems,, Differential Equations & Applications, 5 (2013), 83. doi: 10.7153/dea-05-06. Google Scholar

[35]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar

[36]

W. Zou, Variant fountain theorem and their applivations,, Manuscripta Math., 104 (2001), 343. doi: 10.1007/s002290170032. Google Scholar

show all references

References:
[1]

C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. Google Scholar

[3]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles,, Izv. Akad. Nauk SSSR Ser., 4 (1940), 17. Google Scholar

[4]

H. Brézis, Analyse fonctionelle. Théorie et applications,, Masson, (1983). Google Scholar

[5]

C. Chen, J. Huang and L. Liu, Multiple solutions to the nonhomogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities,, Applied Mathematics Letters, 26 (2013), 754. doi: 10.1016/j.aml.2013.02.011. Google Scholar

[6]

S. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $\mathbbR^N$,, Nonlinear Anal. RWA, 14 (2013), 1477. doi: 10.1016/j.nonrwa.2012.10.010. Google Scholar

[7]

B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, Nonlinear Anal., 71 (2009), 4883. doi: 10.1016/j.na.2009.03.065. Google Scholar

[8]

F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus,, Appl. Math. Lett., 22 (2009), 819. doi: 10.1016/j.aml.2008.06.042. Google Scholar

[9]

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

[10]

A. Fiscella, Saddle point solutions for non-local elliptic operators,, preprint, (2012). Google Scholar

[11]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator,, Nonlinear Analysis, 94 (2014), 156. doi: 10.1016/j.na.2013.08.011. Google Scholar

[12]

X. He and W. Zou, Infinitely many positive solutions of Kirchhoff type problems,, Nonlinear Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar

[13]

X. He and W. Zou, Multiplicity solutions of for a class of Kirchhoff type problems,, Acta Mathematicae Applicatae Sinica, 26 (2010), 387. doi: 10.1007/s10255-010-0005-2. Google Scholar

[14]

Kirchhoff and G. Mechanik, Teubner,, Leipzig, (1883). Google Scholar

[15]

Y. Li, F. Li and J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass,, J. Math. Anal. Appl., 410 (2014), 361. doi: 10.1016/j.jmaa.2013.08.030. Google Scholar

[16]

J. L. Lions, On some quations in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial differential Equations,, Proc. Internat. Sympos., (1977), 284. Google Scholar

[17]

D. Liu and P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation,, Nonlinear Anal., 75 (2012), 5032. doi: 10.1016/j.na.2012.04.018. Google Scholar

[18]

T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar

[19]

A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition,, Nonlinear Anal., 70 (2009), 1275. doi: 10.1016/j.na.2008.02.011. Google Scholar

[20]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar

[21]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb.(N.S.), 168 (1975), 152. Google Scholar

[22]

R. Servadei, The Yamabe equation in a non-local setting,, Advances in Nonlinear Analysis, 2 (2013), 235. doi: 10.1515/anona-2013-0008. Google Scholar

[23]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity,, Contemp. Math., 595 (2013), 317. doi: 10.1090/conm/595/11809. Google Scholar

[24]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, Rev. Mat. Iberoam., 29 (2013), 1091. doi: 10.4171/RMI/750. Google Scholar

[25]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, J. Math. Anal. Appl., 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[26]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete and Continuous Dynamical Systems, 5 (2013), 2105. Google Scholar

[27]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, Commun. Pure Appl. Anal., 12 (2013), 2445. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[29]

R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent,, preprint (2012), (2012), 12. Google Scholar

[30]

G. F. Sun and K. M. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem,, Math. Commun., 19 (2014), 183. Google Scholar

[31]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations,, Nonlinear Analysis, 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061. Google Scholar

[32]

M. Willem, Minimax Theorems,, Birkhauser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[33]

Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent,, Commun. Pure Appl. Anal., 12 (2013), 2773. doi: 10.3934/cpaa.2013.12.2773. Google Scholar

[34]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems,, Differential Equations & Applications, 5 (2013), 83. doi: 10.7153/dea-05-06. Google Scholar

[35]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar

[36]

W. Zou, Variant fountain theorem and their applivations,, Manuscripta Math., 104 (2001), 343. doi: 10.1007/s002290170032. Google Scholar

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