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November  2017, 16(6): 2157-2175. doi: 10.3934/cpaa.2017107

## Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities

 1 School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China 2 School of Mathematics and Information, China West Normal University, Nanchong, Sichuan, 637002, China

* Corresponding author

Received  January 2017 Revised  April 2017 Published  July 2017

Fund Project: supported by National Natural Science Foundation of China(No. 11471267); Natural Science Foundation of Education of Guizhou Province(No. KY[2016]046); Research Foundation of China West Normal University(No. 16E014;No. 15D006).

In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, by the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

Citation: Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107
##### References:
 [1] C. O. Alves, F. J. S. A. Corrȇa 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, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078. [3] 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. [4] 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. [5] B. T. Cheng, Xian Wu and Liu Jun, Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 521-537.  doi: 10.1007/s00030-011-0141-2. [6] Y. B. Deng, S. J. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500-3527.  doi: 10.1016/j.jfa.2015.09.012. [7] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001. [9] 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. [10] G. B. Li and H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011. [11] 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. [12] J. F. Liao, P. Zhang, J. Liu and C. L. 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. F. Liao, P. Zhang, J. Liu and C. L. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1959-1974.  doi: 10.3934/dcdss.2016080. [14] 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. [15] Z. S. Liu and S. J. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747-769.  doi: 10.1007/s00033-014-0431-8. [16] 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. [17] 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. [18] D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914.  doi: 10.1007/s00030-014-0271-4. [19] J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.  doi: 10.1016/j.na.2012.01.004. [20] P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 54 (2015), 1-22.  doi: 10.1007/s00526-015-0883-5. [21] W. Rudin, Real and Complex Analysis McGraw-Hill, New York, London etc. 1966. [22] J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.  doi: 10.1016/j.na.2010.09.061. [23] X. H. Tang and B. T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032. [24] L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.  doi: 10.1007/s12190-012-0536-1. [25] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in $\mathbb{R}^{3}$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.  doi: 10.1016/j.nonrwa.2010.09.023. [26] M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-laplacian, Nonlinearity, 29 (2016), 3186-3205.  doi: 10.1088/0951-7715/29/10/3186. [27] Q. L. Xie, S. W. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890-924.  doi: 10.1016/j.jde.2016.03.028. [28] 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.

show all references

##### References:
 [1] C. O. Alves, F. J. S. A. Corrȇa 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, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078. [3] 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. [4] 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. [5] B. T. Cheng, Xian Wu and Liu Jun, Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 521-537.  doi: 10.1007/s00030-011-0141-2. [6] Y. B. Deng, S. J. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500-3527.  doi: 10.1016/j.jfa.2015.09.012. [7] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001. [9] 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. [10] G. B. Li and H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011. [11] 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. [12] J. F. Liao, P. Zhang, J. Liu and C. L. 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. F. Liao, P. Zhang, J. Liu and C. L. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1959-1974.  doi: 10.3934/dcdss.2016080. [14] 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. [15] Z. S. Liu and S. J. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747-769.  doi: 10.1007/s00033-014-0431-8. [16] 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. [17] 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. [18] D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914.  doi: 10.1007/s00030-014-0271-4. [19] J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.  doi: 10.1016/j.na.2012.01.004. [20] P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 54 (2015), 1-22.  doi: 10.1007/s00526-015-0883-5. [21] W. Rudin, Real and Complex Analysis McGraw-Hill, New York, London etc. 1966. [22] J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.  doi: 10.1016/j.na.2010.09.061. [23] X. H. Tang and B. T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032. [24] L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.  doi: 10.1007/s12190-012-0536-1. [25] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in $\mathbb{R}^{3}$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.  doi: 10.1016/j.nonrwa.2010.09.023. [26] M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-laplacian, Nonlinearity, 29 (2016), 3186-3205.  doi: 10.1088/0951-7715/29/10/3186. [27] Q. L. Xie, S. W. Ma and X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations, 261 (2016), 890-924.  doi: 10.1016/j.jde.2016.03.028. [28] 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.
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