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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, Rairo Modélisation Math. Anal. Numér., 26 (1992), 447-467.
[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.
[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.
[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.
[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.
[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.
[13]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[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.
[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.
[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.
[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.
[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.
[19]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, Rairo Modélisation Math. Anal. Numér., 26 (1992), 447-467.
[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.
[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.
[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.
[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.
[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.
[13]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[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.
[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.
[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.
[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.
[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.
[19]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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