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.

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]

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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