December  2020, 13(12): 3347-3355. doi: 10.3934/dcdss.2020127

Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball

1. 

Department of Mathematics, Harvey Mudd College, Claremont CA 91711, USA

2. 

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received  December 2018 Revised  March 2019 Published  November 2019

Fund Project: Supported by the Science and Technology of Chongqing Educational Commission(Grant No. KJ1600618), the Research Funds of Chongqing Technology and Business University (Grant no.1756001) and China Scholarship Council

We prove the existence of infinitely many radial solutions to a Kirchhoff type problem in a ball with a super-cubic nonlinearity. Our methods rely on bifurcation analysis and energy estimates.

Citation: Alfonso Castro, Shu-Zhi Song. Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3347-3355. doi: 10.3934/dcdss.2020127
References:
[1]

P. Chen and X. H. Tang, Existence and multiplicity results for infinitely many solutions for Kirchhoff-type problems in $\mathbb{R}^N$, Math. Methods Appl. Sci., 37 (2014), 1828-1837.  doi: 10.1002/mma.2938.  Google Scholar

[2]

B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbb{R}^N$, J. Nonlinear Sci. Appl., 9 (2016), 652-660.  doi: 10.22436/jnsa.009.02.28.  Google Scholar

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

L. Duan and L. H. Huang, Infinitely many solutions for sublinear Schrödinger-Kirchhoff-type equations with general potentials, Results Math., 66 (2014), 181-197.  doi: 10.1007/s00025-014-0371-9.  Google Scholar

[6]

W. J. Feng and X. J. Feng, Multiple solutions for Kirchhoff equations under the partially sublinear case, J. Funct. Spaces, (2015), Art. ID 610858, 4 pp. doi: 10.1155/2015/610858.  Google Scholar

[7]

Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 428 (2015), 1054-1069.  doi: 10.1016/j.jmaa.2015.03.064.  Google Scholar

[8]

X.-M. He and W.-M. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 387-394.  doi: 10.1007/s10255-010-0005-2.  Google Scholar

[9]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb{R}^N$, J. Math. Anal. Appl., 369 (2010), 564-574.  doi: 10.1016/j.jmaa.2010.03.059.  Google Scholar

[10]

A. Li and J. B. Su, Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in $\mathbb{R}^3$, Z. Angew. Math. Phys., 66 (2015), 3147-3158.  doi: 10.1007/s00033-015-0551-9.  Google Scholar

[11]

L. Li and X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 435 (2016), 955-967.  doi: 10.1016/j.jmaa.2015.10.075.  Google Scholar

[12]

J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 417 (2014), 65-79.  doi: 10.1016/j.jmaa.2014.03.027.  Google Scholar

[13]

J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.  doi: 10.1016/j.na.2012.01.004.  Google Scholar

[14]

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

[15]

S. Z. SongS. J. Chen and C. L. Tang, Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues, Discrete Contin. Dyn. Syst., 36 (2016), 6453-6473.  doi: 10.3934/dcds.2016078.  Google Scholar

[16]

S.-Z. Song, C. L. Tang and S.-J. Chen, Multiple solutions for Kirchhoff type problem near resonance, Electron. J. Differential Equations, 2015, (2015), 7 pp.  Google Scholar

[17]

J. J. SunL. LiM. Cencelj and B. Gabrovšek, Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^3$, Nonlinear Analysis, 186 (2019), 33-54.  doi: 10.1016/j.na.2018.10.007.  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]

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

[20]

Q. L. XieS. W. Ma and X. Zhang, Infinitely many bound state solutions of Kirchhoff problem in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 29 (2016), 80-97.  doi: 10.1016/j.nonrwa.2015.10.010.  Google Scholar

[21]

X. Z. Yao and C. L. Mu, Infinitely many sign-changing solutions for Kirchhoff-type equations with power nonlinearity, Electron. J. Differential Equations, 2016 (2016), 7 pp.  Google Scholar

[22]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differ. Equ. Appl., 5 (2013), 83-92.  doi: 10.7153/dea-05-06.  Google Scholar

[23]

Y. W. Ye and C. L. Tang, Multiple solutions for Kirchhoff-type equations in $\mathbb{R}^N$, J. Math. Phys., 54 (2013), 081508, 16 pp. doi: 10.1063/1.4819249.  Google Scholar

[24]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

[25]

Q. Y. Zhang and B. Xu, Infinitely many solutions for Schrödinger-Kirchhoff-type equations involving indefinite potential, Electron. J. Qual. Theory Differ. Equ., 2017, (2017), 17 pp. doi: 10.14232/ejqtde.2017.1.58.  Google Scholar

show all references

References:
[1]

P. Chen and X. H. Tang, Existence and multiplicity results for infinitely many solutions for Kirchhoff-type problems in $\mathbb{R}^N$, Math. Methods Appl. Sci., 37 (2014), 1828-1837.  doi: 10.1002/mma.2938.  Google Scholar

[2]

B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbb{R}^N$, J. Nonlinear Sci. Appl., 9 (2016), 652-660.  doi: 10.22436/jnsa.009.02.28.  Google Scholar

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

L. Duan and L. H. Huang, Infinitely many solutions for sublinear Schrödinger-Kirchhoff-type equations with general potentials, Results Math., 66 (2014), 181-197.  doi: 10.1007/s00025-014-0371-9.  Google Scholar

[6]

W. J. Feng and X. J. Feng, Multiple solutions for Kirchhoff equations under the partially sublinear case, J. Funct. Spaces, (2015), Art. ID 610858, 4 pp. doi: 10.1155/2015/610858.  Google Scholar

[7]

Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 428 (2015), 1054-1069.  doi: 10.1016/j.jmaa.2015.03.064.  Google Scholar

[8]

X.-M. He and W.-M. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 387-394.  doi: 10.1007/s10255-010-0005-2.  Google Scholar

[9]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb{R}^N$, J. Math. Anal. Appl., 369 (2010), 564-574.  doi: 10.1016/j.jmaa.2010.03.059.  Google Scholar

[10]

A. Li and J. B. Su, Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in $\mathbb{R}^3$, Z. Angew. Math. Phys., 66 (2015), 3147-3158.  doi: 10.1007/s00033-015-0551-9.  Google Scholar

[11]

L. Li and X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 435 (2016), 955-967.  doi: 10.1016/j.jmaa.2015.10.075.  Google Scholar

[12]

J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 417 (2014), 65-79.  doi: 10.1016/j.jmaa.2014.03.027.  Google Scholar

[13]

J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.  doi: 10.1016/j.na.2012.01.004.  Google Scholar

[14]

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

[15]

S. Z. SongS. J. Chen and C. L. Tang, Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues, Discrete Contin. Dyn. Syst., 36 (2016), 6453-6473.  doi: 10.3934/dcds.2016078.  Google Scholar

[16]

S.-Z. Song, C. L. Tang and S.-J. Chen, Multiple solutions for Kirchhoff type problem near resonance, Electron. J. Differential Equations, 2015, (2015), 7 pp.  Google Scholar

[17]

J. J. SunL. LiM. Cencelj and B. Gabrovšek, Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^3$, Nonlinear Analysis, 186 (2019), 33-54.  doi: 10.1016/j.na.2018.10.007.  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]

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

[20]

Q. L. XieS. W. Ma and X. Zhang, Infinitely many bound state solutions of Kirchhoff problem in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 29 (2016), 80-97.  doi: 10.1016/j.nonrwa.2015.10.010.  Google Scholar

[21]

X. Z. Yao and C. L. Mu, Infinitely many sign-changing solutions for Kirchhoff-type equations with power nonlinearity, Electron. J. Differential Equations, 2016 (2016), 7 pp.  Google Scholar

[22]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differ. Equ. Appl., 5 (2013), 83-92.  doi: 10.7153/dea-05-06.  Google Scholar

[23]

Y. W. Ye and C. L. Tang, Multiple solutions for Kirchhoff-type equations in $\mathbb{R}^N$, J. Math. Phys., 54 (2013), 081508, 16 pp. doi: 10.1063/1.4819249.  Google Scholar

[24]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

[25]

Q. Y. Zhang and B. Xu, Infinitely many solutions for Schrödinger-Kirchhoff-type equations involving indefinite potential, Electron. J. Qual. Theory Differ. Equ., 2017, (2017), 17 pp. doi: 10.14232/ejqtde.2017.1.58.  Google Scholar

Figure 1.  Bifurcation curves for solutions to (1.6)
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