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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 and 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), 85-93. doi: 10.1016/j.camwa.2005.01.008.

[2]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[3]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles, Izv. Akad. Nauk SSSR Ser., 4 (1940), 17-26.

[4]

H. Brézis, Analyse fonctionelle. Théorie et applications, Masson, Paris, 1983.

[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), 754-759. doi: 10.1016/j.aml.2013.02.011.

[6]

S. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $\mathbb{R}^N2$, Nonlinear Anal. RWA, 14 (2013), 1477-1486. doi: 10.1016/j.nonrwa.2012.10.010.

[7]

B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065.

[8]

F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 22 (2009), 819-822. doi: 10.1016/j.aml.2008.06.042.

[9]

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.

[10]

A. Fiscella, Saddle point solutions for non-local elliptic operators, preprint, (2012), available at http://arxiv.org/abs/1210.8401.

[11]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Analysis, 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.

[12]

X. He and W. Zou, Infinitely many positive solutions of Kirchhoff type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.

[13]

X. He and W. Zou, Multiplicity solutions of for a class of Kirchhoff type problems, Acta Mathematicae Applicatae Sinica, English Series, 26 (2010), 387-394. doi: 10.1007/s10255-010-0005-2.

[14]

Kirchhoff and G. Mechanik, Teubner, Leipzig, 1883.

[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), 361-374. doi: 10.1016/j.jmaa.2013.08.030.

[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., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol.30, North- Holland, Amsterdam, 1978, 284-346.

[17]

D. Liu and P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation, Nonlinear Anal., 75 (2012), 5032-5038. doi: 10.1016/j.na.2012.04.018.

[18]

T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1.

[19]

A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.

[20]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.

[21]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb.(N.S.), 168 (1975), 152-166.

[22]

R. Servadei, The Yamabe equation in a non-local setting, Advances in Nonlinear Analysis, 2 (2013), 235-270. doi: 10.1515/anona-2013-0008.

[23]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.

[24]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[25]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[26]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete and Continuous Dynamical Systems, 5 (2013), 2105-2137.

[27]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4.

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[29]

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

[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), 183-194.

[31]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Analysis, 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.

[32]

M. Willem, Minimax Theorems, Birkhauser, Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[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), 2773-2786. doi: 10.3934/cpaa.2013.12.2773.

[34]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differential Equations & Applications, 5 (2013), 83-92. doi: 10.7153/dea-05-06.

[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), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

[36]

W. Zou, Variant fountain theorem and their applivations, Manuscripta Math., 104 (2001), 343-358. doi: 10.1007/s002290170032.

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), 85-93. doi: 10.1016/j.camwa.2005.01.008.

[2]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[3]

S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles, Izv. Akad. Nauk SSSR Ser., 4 (1940), 17-26.

[4]

H. Brézis, Analyse fonctionelle. Théorie et applications, Masson, Paris, 1983.

[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), 754-759. doi: 10.1016/j.aml.2013.02.011.

[6]

S. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $\mathbb{R}^N2$, Nonlinear Anal. RWA, 14 (2013), 1477-1486. doi: 10.1016/j.nonrwa.2012.10.010.

[7]

B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065.

[8]

F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 22 (2009), 819-822. doi: 10.1016/j.aml.2008.06.042.

[9]

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.

[10]

A. Fiscella, Saddle point solutions for non-local elliptic operators, preprint, (2012), available at http://arxiv.org/abs/1210.8401.

[11]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Analysis, 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.

[12]

X. He and W. Zou, Infinitely many positive solutions of Kirchhoff type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021.

[13]

X. He and W. Zou, Multiplicity solutions of for a class of Kirchhoff type problems, Acta Mathematicae Applicatae Sinica, English Series, 26 (2010), 387-394. doi: 10.1007/s10255-010-0005-2.

[14]

Kirchhoff and G. Mechanik, Teubner, Leipzig, 1883.

[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), 361-374. doi: 10.1016/j.jmaa.2013.08.030.

[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., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol.30, North- Holland, Amsterdam, 1978, 284-346.

[17]

D. Liu and P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation, Nonlinear Anal., 75 (2012), 5032-5038. doi: 10.1016/j.na.2012.04.018.

[18]

T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1.

[19]

A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.

[20]

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.

[21]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb.(N.S.), 168 (1975), 152-166.

[22]

R. Servadei, The Yamabe equation in a non-local setting, Advances in Nonlinear Analysis, 2 (2013), 235-270. doi: 10.1515/anona-2013-0008.

[23]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.

[24]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[25]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[26]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete and Continuous Dynamical Systems, 5 (2013), 2105-2137.

[27]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4.

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[29]

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

[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), 183-194.

[31]

J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Analysis, 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.

[32]

M. Willem, Minimax Theorems, Birkhauser, Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[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), 2773-2786. doi: 10.3934/cpaa.2013.12.2773.

[34]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differential Equations & Applications, 5 (2013), 83-92. doi: 10.7153/dea-05-06.

[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), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

[36]

W. Zou, Variant fountain theorem and their applivations, Manuscripta Math., 104 (2001), 343-358. doi: 10.1007/s002290170032.

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