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