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January  2022, 42(1): 1-19. doi: 10.3934/dcds.2021106

On the compactness threshold in the critical Kirchhoff equation

Erisa Hasani and  Kanishka Perera ,

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

*Corresponding author: Kanishka Perera

Received  December 2020 Revised  May 2021 Published  January 2022 Early access  July 2021

We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.

Keywords: Critical Kirchhoff equation, general nonlocal term, compactness threshold, existence, multiplicity.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35B33, 35J20.
Citation: Erisa Hasani, Kanishka Perera. On the compactness threshold in the critical Kirchhoff equation. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 1-19. doi: 10.3934/dcds.2021106
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[3]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[4]

H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.  doi: 10.1002/cpa.3160440808.  Google Scholar

[5]

P. Drábek and Y. X. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in RN with critical Sobolev exponent, J. Differential Equations, 140 (1997), 106-132.  doi: 10.1006/jdeq.1997.3306.  Google Scholar

[6]

E. Hebey, Compactness and the Palais-Smale property for critical Kirchhoff equations in closed manifolds, Pacific J. Math., 280 (2016), 41-50.   Google Scholar

[7]

Y. Huang, Z. Liu and Y. Wu, On Kirchhoff type equations with critical Sobolev exponent, J. Math. Anal. Appl., 462 (2018), 483-504.  doi: 10.1016/j.jmaa.2018.02.023.  Google Scholar

[8]

S. J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6-32.  doi: 10.1006/jmaa.1995.1002.  Google Scholar

[9]

J.-F. Liao, X.-F. Ke, J. Liu and C.-L. Tang, The Brezis-Nirenberg result for the Kirchhoff-type equation in dimension four, Appl. Anal., 97 (2018), 2720-2726.  doi: 10.1080/00036811.2017.1387248.  Google Scholar

[10]

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.  doi: 10.1016/j.jde.2014.05.002.  Google Scholar

[11]

D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914.  doi: 10.1007/s00030-014-0271-4.  Google Scholar

[12]

D. Naimen and M. Shibata, Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity in high dimension, Nonlinear Anal., 186 (2019), 187-208.  doi: 10.1016/j.na.2019.02.003.  Google Scholar

[13]

K. Perera and Z. 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

[14]

Q.-L. Xie, X.-P. Wu and C.-L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786.  doi: 10.3934/cpaa.2013.12.2773.  Google Scholar

[15]

X. Yao and C. Mu, Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension, Math. Methods Appl. Sci., 39 (2016), 3722-3734.  doi: 10.1002/mma.3821.  Google Scholar

[16]

C. Zhang and Z. Liu, Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem, Appl. Math. Lett., 69 (2017), 87-93.  doi: 10.1016/j.aml.2017.01.016.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[3]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[4]

H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.  doi: 10.1002/cpa.3160440808.  Google Scholar

[5]

P. Drábek and Y. X. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in RN with critical Sobolev exponent, J. Differential Equations, 140 (1997), 106-132.  doi: 10.1006/jdeq.1997.3306.  Google Scholar

[6]

E. Hebey, Compactness and the Palais-Smale property for critical Kirchhoff equations in closed manifolds, Pacific J. Math., 280 (2016), 41-50.   Google Scholar

[7]

Y. Huang, Z. Liu and Y. Wu, On Kirchhoff type equations with critical Sobolev exponent, J. Math. Anal. Appl., 462 (2018), 483-504.  doi: 10.1016/j.jmaa.2018.02.023.  Google Scholar

[8]

S. J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6-32.  doi: 10.1006/jmaa.1995.1002.  Google Scholar

[9]

J.-F. Liao, X.-F. Ke, J. Liu and C.-L. Tang, The Brezis-Nirenberg result for the Kirchhoff-type equation in dimension four, Appl. Anal., 97 (2018), 2720-2726.  doi: 10.1080/00036811.2017.1387248.  Google Scholar

[10]

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193.  doi: 10.1016/j.jde.2014.05.002.  Google Scholar

[11]

D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 885-914.  doi: 10.1007/s00030-014-0271-4.  Google Scholar

[12]

D. Naimen and M. Shibata, Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity in high dimension, Nonlinear Anal., 186 (2019), 187-208.  doi: 10.1016/j.na.2019.02.003.  Google Scholar

[13]

K. Perera and Z. 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

[14]

Q.-L. Xie, X.-P. Wu and C.-L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786.  doi: 10.3934/cpaa.2013.12.2773.  Google Scholar

[15]

X. Yao and C. Mu, Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension, Math. Methods Appl. Sci., 39 (2016), 3722-3734.  doi: 10.1002/mma.3821.  Google Scholar

[16]

C. Zhang and Z. Liu, Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem, Appl. Math. Lett., 69 (2017), 87-93.  doi: 10.1016/j.aml.2017.01.016.  Google Scholar

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