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May  2016, 21(3): 883-908. doi: 10.3934/dcdsb.2016.21.883

Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions

Yuanxiao Li 1, , Ming Mei 2, and  Kaijun Zhang 3,

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2. 

Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2
3. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, MO 130024

Received  April 2015 Revised  December 2015 Published  January 2016

This paper deals with a $p$-Kirchhoff type problem involving sign-changing weight functions. It is shown that under certain conditions, by means of variational methods, the existence of multiple nontrivial nonnegative solutions for the problem with the subcritical exponent are obtained. Moreover, in the case of critical exponent, we establish the existence of the solutions and prove that the elliptic equation possesses at least one nontrivial nonnegative solution.
Keywords: $p$-Kirchhoff type equation, truncation argument., Ekeland variational principle, Nehari manifold, sign-changing weight functions.
Mathematics Subject Classification: 35D30, 35J20, 35J6.
Citation: Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
References:
[1]

C. O. Alves and F. J. S. A. Corrêa, On existence of solutions for a class of the problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8 (2001), 43-56.  Google Scholar

[2]

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.  Google Scholar

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K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function, J. Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

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[7]

M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, Rairo Modélisation Math. Anal. Numér., 26 (1992), 447-467.  Google Scholar

[8]

C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar

[9]

B. T. 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.  Google Scholar

[10]

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.  Google Scholar

[11]

F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277. doi: 10.1017/S000497270003570X.  Google Scholar

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P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar

[13]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[14]

G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713. doi: 10.1016/j.jmaa.2012.12.053.  Google Scholar

[15]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[16]

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

[17]

J. C. Huang, C. S. Chen and Z. H. Xiu, Existence and multiplicity results for a $p$-Kirchhoff equation with a concave-convex term, Appl. Math. Lett., 26 (2013), 1070-1075. doi: 10.1016/j.aml.2013.06.001.  Google Scholar

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A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for $p$-Kirchhoff type problems with critical exponent, Electronic J. Differential Equations, 2011 (2011), 1-8.  Google Scholar

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G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

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J. L. Lions, On some questions in boundary value problems of mathmatical physics, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, in North-Holland Math. Stud., North-Holland, Amsterdam, New York, 30 (1978), 284-346.  Google Scholar

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P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case Part I Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[22]

D. C. Liu, On a p-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302-308. doi: 10.1016/j.na.2009.06.052.  Google Scholar

[23]

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

[24]

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.  Google Scholar

[25]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. doi: 10.1016/j.na.2005.03.021.  Google Scholar

[26]

A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 295-298. doi: 10.1016/j.crma.2014.01.015.  Google Scholar

[27]

K. Perera and Z. 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.  Google Scholar

[28]

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.  Google Scholar

show all references

References:
[1]

C. O. Alves and F. J. S. A. Corrêa, On existence of solutions for a class of the problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8 (2001), 43-56.  Google Scholar

[2]

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.  Google Scholar

[3]

C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 2 (2010), 409-417. doi: 10.7153/dea-02-25.  Google Scholar

[4]

G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251. doi: 10.1016/j.jmaa.2010.07.019.  Google Scholar

[5]

K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function, J. Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[6]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7.  Google Scholar

[7]

M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, Rairo Modélisation Math. Anal. Numér., 26 (1992), 447-467.  Google Scholar

[8]

C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar

[9]

B. T. 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.  Google Scholar

[10]

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.  Google Scholar

[11]

F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277. doi: 10.1017/S000497270003570X.  Google Scholar

[12]

P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar

[13]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[14]

G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713. doi: 10.1016/j.jmaa.2012.12.053.  Google Scholar

[15]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar

[16]

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

[17]

J. C. Huang, C. S. Chen and Z. H. Xiu, Existence and multiplicity results for a $p$-Kirchhoff equation with a concave-convex term, Appl. Math. Lett., 26 (2013), 1070-1075. doi: 10.1016/j.aml.2013.06.001.  Google Scholar

[18]

A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for $p$-Kirchhoff type problems with critical exponent, Electronic J. Differential Equations, 2011 (2011), 1-8.  Google Scholar

[19]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[20]

J. L. Lions, On some questions in boundary value problems of mathmatical physics, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, in North-Holland Math. Stud., North-Holland, Amsterdam, New York, 30 (1978), 284-346.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case Part I Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[22]

D. C. Liu, On a p-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302-308. doi: 10.1016/j.na.2009.06.052.  Google Scholar

[23]

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

[24]

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.  Google Scholar

[25]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. doi: 10.1016/j.na.2005.03.021.  Google Scholar

[26]

A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity, C. R. Acad. Sci. Paris, Ser. I, 352 (2014), 295-298. doi: 10.1016/j.crma.2014.01.015.  Google Scholar

[27]

K. Perera and Z. 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.  Google Scholar

[28]

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.  Google Scholar

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