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 and 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 $\mathbb{R}^N2$, Nonlinear Analysis, 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017.

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[3]

G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8.

[4]

Y. Huang and Z. Liu, On a class of Kirchhoff type problems, Arch. Math., 102 (2014), 127-139. doi: 10.1007/s00013-014-0618-4.

[5]

N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials, Discrete Contin. Dyn. Syst., 35 (2015), 943-966. doi: 10.3934/dcds.2015.35.943.

[6]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N2$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[7]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.

[8]

G. Li and H. 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.

[9]

Z. Liu and S. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations, Math. Meth. Appl. Sci., 37 (2014), 571-580. doi: 10.1002/mma.2815.

[10]

Y. Li, F. Li and J. 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.

[11]

J. 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.

[12]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

[13]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

[14]

Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in $\mathbb{R}^N2$, J. Math. Anal. Appl., 425 (2015), 548-564. doi: 10.1016/j.jmaa.2014.12.017.

show all references

References:
[1]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbb{R}^N2$, Nonlinear Analysis, 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017.

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[3]

G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8.

[4]

Y. Huang and Z. Liu, On a class of Kirchhoff type problems, Arch. Math., 102 (2014), 127-139. doi: 10.1007/s00013-014-0618-4.

[5]

N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials, Discrete Contin. Dyn. Syst., 35 (2015), 943-966. doi: 10.3934/dcds.2015.35.943.

[6]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N2$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[7]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.

[8]

G. Li and H. 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.

[9]

Z. Liu and S. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations, Math. Meth. Appl. Sci., 37 (2014), 571-580. doi: 10.1002/mma.2815.

[10]

Y. Li, F. Li and J. 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.

[11]

J. 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.

[12]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

[13]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

[14]

Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in $\mathbb{R}^N2$, J. Math. Anal. Appl., 425 (2015), 548-564. doi: 10.1016/j.jmaa.2014.12.017.

[1]

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

[2]

Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135

[3]

Wei-Xi Li, Lvqiao Liu. Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off. Kinetic and Related Models, 2020, 13 (5) : 1029-1046. doi: 10.3934/krm.2020036

[4]

Die Hu, Xianhua Tang, Qi Zhang. Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1071-1091. doi: 10.3934/cpaa.2022010

[5]

Zhi-Guo Wu, Wen Guan, Da-Bin Wang. Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2495-2528. doi: 10.3934/cpaa.2022058

[6]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287

[7]

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

[8]

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

[9]

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 and Applied Analysis, 2020, 19 (2) : 941-966. doi: 10.3934/cpaa.2020043

[10]

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

[11]

Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034

[12]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1301-1322. doi: 10.3934/dcdsb.2021091

[13]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1927-1954. doi: 10.3934/dcdsb.2021115

[14]

He Zhang, Haibo Chen. The effect of the weight function on the number of nodal solutions of the Kirchhoff-type equations in high dimensions. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2701-2721. doi: 10.3934/cpaa.2022069

[15]

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

[16]

Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2095-2107. doi: 10.3934/dcdss.2020160

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (199)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]