# American Institute of Mathematical Sciences

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 $\mathbbR^N$, Nonlinear Analysis, 75 (2012), 2750-2759. 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, Arch. Rational Mech. Anal., 82 (1983), 313-345. 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, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. 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, Adv. Differential Equations, 11 (2006), 813-840.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] M. Willem, Minimax Theorems, Birkhäuser, 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 $\mathbbR^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar [14] Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in $\mathbbR^N$, J. Math. Anal. Appl., 425 (2015), 548-564. doi: 10.1016/j.jmaa.2014.12.017.  Google Scholar

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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.  Google Scholar [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.  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, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. 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, Adv. Differential Equations, 11 (2006), 813-840.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] M. Willem, Minimax Theorems, Birkhäuser, 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 $\mathbbR^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar [14] Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in $\mathbbR^N$, J. Math. Anal. Appl., 425 (2015), 548-564. doi: 10.1016/j.jmaa.2014.12.017.  Google Scholar
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