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

Jiu  Liu; Jia-Feng  Liao; Chun-Lei Tang

doi:10.3934/cpaa.2016.15.445

Article Contents

2016, Volume 15, Issue 2: 445-455. Doi: 10.3934/cpaa.2016.15.445

This issue Previous Article Schrödinger-Kirchhoff-Poisson type systems Next Article Competing interactions and traveling wave solutions in lattice differential equations

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

1.
School of Mathematics and Statistics, Southwest University, Chongqing, 400715
2.
School of Mathematics and Statistics, Southwest University, Chongqing 400715

Received: February 28, 2015

Revised: October 31, 2015

Published: February 2016

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Abstract

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.

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Mathematics Subject Classification: 35B09, 35B38, 35J20.

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References

[1]	C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N$, 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 $\mathbbR^N$, 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 $\mathbbR^3$, 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 $\mathbbR^N$, 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 $\mathbbR^N$, J. Math. Anal. Appl., 425 (2015), 548-564.doi: 10.1016/j.jmaa.2014.12.017.