# American Institute of Mathematical Sciences

December  2016, 9(6): 1959-1974. doi: 10.3934/dcdss.2016080

## 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

Received  May 2015 Revised  September 2016 Published  November 2016

In this paper, we study a class of Kirchhoff type problems with resonance \begin{equation*} \begin{cases} -\left(a+b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\nu u^{3}+ \lambda |u|^{q-1}u,&\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on} \ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^{3}$ is a bounded domain, $a,b,\nu,\lambda>0$ and $0< q <1$. By a minimizing method, we obtain the existence of positive ground state solutions for all $0<\nu\leq b\nu_{1}$ and $\lambda>0$. Furthermore, using the Nehari method, we obtain two positive solutions for all $\nu>b\nu_{1}$ and $0<\lambda<\tilde{\lambda},$ where $\nu_{1}$ is the first eigenvalue of problem (5) and $\tilde{\lambda}$ is a positive constant. And one of the two positive solutions is a ground state solution.
Citation: Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080
##### 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] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. [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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##### 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] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. [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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