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Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance
1. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China |
2. | School of Mathematics and Computational Science, Zunyi Normal College, Zunyi 563002, China |
3. | School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Duyun 558000, China |
References:
[1] |
A. Ambrosetti, H. Brézis and G. Cermi, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems, J. Math. Anal. Appl., 373 (2011), 248-251.
doi: 10.1016/j.jmaa.2010.07.019. |
[3] |
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. |
[4] |
N. Daisuke, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.
doi: 10.1016/j.jde.2014.05.002. |
[5] |
X. M. He and W. M. Zou, Infnitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.
doi: 10.1016/j.na.2008.02.021. |
[6] |
X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[7] | |
[8] |
C. Y. Lei, J. F. Liao and C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538.
doi: 10.1016/j.jmaa.2014.07.031. |
[9] |
Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[10] |
Z. P. Liang, F. Y. Li and J. P. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155-167.
doi: 10.1016/j.anihpc.2013.01.006. |
[11] |
J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24-30.
doi: 10.1016/j.aml.2016.03.001. |
[12] |
J. F. Liao, P. Zhang, J. Liu and C. T. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148.
doi: 10.1016/j.jmaa.2015.05.038. |
[13] |
J. Lions, On some questions 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, New York, 1978, pp. 284-346. |
[14] |
X. Liu and Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721-733.
doi: 10.3934/cpaa.2013.12.721. |
[15] |
A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243.
doi: 10.1016/j.jmaa.2011.05.021. |
[16] |
A. M. Mao and Z. T. 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. |
[17] |
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. |
[18] |
J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[19] |
J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33 (2013), 2139-2154.
doi: 10.3934/dcds.2013.33.2139. |
[20] |
J. T. Sun and T. F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792.
doi: 10.1016/j.jde.2013.12.006. |
[21] |
M. Willem, Minimax Theorems, Birthäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[22] |
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. |
[23] |
H. Zhang and F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671-1692.
doi: 10.1016/j.jmaa.2014.10.062. |
show all references
References:
[1] |
A. Ambrosetti, H. Brézis and G. Cermi, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems, J. Math. Anal. Appl., 373 (2011), 248-251.
doi: 10.1016/j.jmaa.2010.07.019. |
[3] |
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. |
[4] |
N. Daisuke, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.
doi: 10.1016/j.jde.2014.05.002. |
[5] |
X. M. He and W. M. Zou, Infnitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.
doi: 10.1016/j.na.2008.02.021. |
[6] |
X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[7] | |
[8] |
C. Y. Lei, J. F. Liao and C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538.
doi: 10.1016/j.jmaa.2014.07.031. |
[9] |
Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[10] |
Z. P. Liang, F. Y. Li and J. P. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155-167.
doi: 10.1016/j.anihpc.2013.01.006. |
[11] |
J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24-30.
doi: 10.1016/j.aml.2016.03.001. |
[12] |
J. F. Liao, P. Zhang, J. Liu and C. T. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148.
doi: 10.1016/j.jmaa.2015.05.038. |
[13] |
J. Lions, On some questions 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, New York, 1978, pp. 284-346. |
[14] |
X. Liu and Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721-733.
doi: 10.3934/cpaa.2013.12.721. |
[15] |
A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243.
doi: 10.1016/j.jmaa.2011.05.021. |
[16] |
A. M. Mao and Z. T. 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. |
[17] |
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. |
[18] |
J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[19] |
J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33 (2013), 2139-2154.
doi: 10.3934/dcds.2013.33.2139. |
[20] |
J. T. Sun and T. F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792.
doi: 10.1016/j.jde.2013.12.006. |
[21] |
M. Willem, Minimax Theorems, Birthäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[22] |
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. |
[23] |
H. Zhang and F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671-1692.
doi: 10.1016/j.jmaa.2014.10.062. |
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