March  2016, 15(2): 445-455. doi: 10.3934/cpaa.2016.15.445

Positive solution for the Kirchhoff-type equations involving general subcritical growth

1. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China, China

2. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715

Received  March 2015 Revised  November 2015 Published  January 2016

In this paper, the existence of a positive solution for the Kirchhoff-type equations in $\mathbb{R}^N$ is proved by using cut-off and monotonicity tricks, which unify and sharply improve the results of Li et al. [Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012) 2285--2294]. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear at infinity.
Citation: 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
References:
[1]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N$,, \emph{Nonlinear Analysis}, 75 (2012), 2750. doi: 10.1016/j.na.2011.11.017. Google Scholar

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar

[3]

G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities,, \emph{Arch. Rational Mech. Anal.}, 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[4]

Y. Huang and Z. Liu, On a class of Kirchhoff type problems,, \emph{Arch. Math.}, 102 (2014), 127. doi: 10.1007/s00013-014-0618-4. Google Scholar

[5]

N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials,, \emph{Discrete Contin. Dyn. Syst.}, 35 (2015), 943. doi: 10.3934/dcds.2015.35.943. Google Scholar

[6]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbbR^N$,, \emph{Proc. Roy. Soc. Edinburgh}, 129 (1999), 787. doi: 10.1017/S0308210500013147. Google Scholar

[7]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 11 (2006), 813. Google Scholar

[8]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbbR^3$,, \emph{J. Differential Equations}, 257 (2014), 566. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[9]

Z. Liu and S. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations,, \emph{Math. Meth. Appl. Sci.}, 37 (2014), 571. doi: 10.1002/mma.2815. Google Scholar

[10]

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differential Equations}, 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017. Google Scholar

[11]

J. Sun and T.-F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well,, \emph{J. Differential Equations}, 256 (2014), 1771. doi: 10.1016/j.jde.2013.12.006. Google Scholar

[12]

M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[13]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[14]

Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 425 (2015), 548. doi: 10.1016/j.jmaa.2014.12.017. Google Scholar

show all references

References:
[1]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N$,, \emph{Nonlinear Analysis}, 75 (2012), 2750. doi: 10.1016/j.na.2011.11.017. Google Scholar

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar

[3]

G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities,, \emph{Arch. Rational Mech. Anal.}, 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[4]

Y. Huang and Z. Liu, On a class of Kirchhoff type problems,, \emph{Arch. Math.}, 102 (2014), 127. doi: 10.1007/s00013-014-0618-4. Google Scholar

[5]

N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials,, \emph{Discrete Contin. Dyn. Syst.}, 35 (2015), 943. doi: 10.3934/dcds.2015.35.943. Google Scholar

[6]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbbR^N$,, \emph{Proc. Roy. Soc. Edinburgh}, 129 (1999), 787. doi: 10.1017/S0308210500013147. Google Scholar

[7]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 11 (2006), 813. Google Scholar

[8]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbbR^3$,, \emph{J. Differential Equations}, 257 (2014), 566. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[9]

Z. Liu and S. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations,, \emph{Math. Meth. Appl. Sci.}, 37 (2014), 571. doi: 10.1002/mma.2815. Google Scholar

[10]

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differential Equations}, 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017. Google Scholar

[11]

J. Sun and T.-F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well,, \emph{J. Differential Equations}, 256 (2014), 1771. doi: 10.1016/j.jde.2013.12.006. Google Scholar

[12]

M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[13]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[14]

Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 425 (2015), 548. doi: 10.1016/j.jmaa.2014.12.017. Google Scholar

[1]

Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic & Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583

[2]

Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124

[3]

Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032

[4]

Sami Aouaoui. A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1351-1370. doi: 10.3934/cpaa.2016.15.1351

[5]

Zhiqing Liu, Zhong Bo Fang. Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term. Communications on Pure & Applied Analysis, 2020, 19 (2) : 941-966. doi: 10.3934/cpaa.2020043

[6]

Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137

[7]

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

[8]

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

[9]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731

[10]

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

[11]

Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010

[12]

F. D. Araruna, F. O. Matias, M. P. Matos, S. M. S. Souza. Hidden regularity for the Kirchhoff equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1049-1056. doi: 10.3934/cpaa.2008.7.1049

[13]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154

[14]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214

[15]

Takahiro Hashimoto. Pohozaev-Ôtani type inequalities for weak solutions of quasilinear elliptic equations with homogeneous coefficients. Conference Publications, 2011, 2011 (Special) : 643-652. doi: 10.3934/proc.2011.2011.643

[16]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257

[17]

Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91

[18]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111

[19]

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

[20]

Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]