American Institute of Mathematical Sciences

Advanced
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

Jiafeng Liao 1, , Peng Zhang 2, , Jiu Liu 3, and  Chunlei Tang 1,

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.
Keywords: positive solution, Nehari method., resonance, Kirchhoff type equation.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35B09, 35A1.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857
[2]

Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289
[3]

Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139
[4]

Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure & Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445
[5]

To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694
[6]

Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078
[7]

Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721
[8]

Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 229-243. doi: 10.3934/dcdss.2021029
[9]

José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653
[10]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
[11]

Ling Ding, Shu-Ming Sun. Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1663-1685. doi: 10.3934/dcdss.2016069
[12]

Zhijian Yang, Na Feng, Yanan Li. Robust attractors for a Kirchhoff-Boussinesq type equation. Evolution Equations & Control Theory, 2020, 9 (2) : 469-486. doi: 10.3934/eect.2020020
[13]

Min Liu, Zhongwei Tang. Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3365-3398. doi: 10.3934/dcds.2019139
[14]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089
[15]

Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations & Control Theory, 2021, 10 (4) : 723-732. doi: 10.3934/eect.2020088
[16]

Die Hu, Xianhua Tang, Qi Zhang. Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022010
[17]

Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015
[18]

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 & Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107
[19]

Jie Yang, Haibo Chen. Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3065-3092. doi: 10.3934/cpaa.2021096
[20]

Weipeng Hu, Zichen Deng, Yuyue Qin. Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation. Journal of Geometric Mechanics, 2013, 5 (3) : 295-318. doi: 10.3934/jgm.2013.5.295

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (181)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy