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

On the compactness threshold in the critical Kirchhoff equation

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.

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. HuangZ. 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. LiaoX.-F. KeJ. 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. XieX.-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. HuangZ. 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. LiaoX.-F. KeJ. 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. XieX.-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

[1]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[2]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015

[3]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

[4]

Wenjun Liu, Biqing Zhu, Gang Li, Danhua Wang. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations & Control Theory, 2017, 6 (2) : 239-260. doi: 10.3934/eect.2017013

[5]

Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729

[6]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[7]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016

[8]

Mohammad Al-Gharabli, Mohamed Balegh, Baowei Feng, Zayd Hajjej, Salim A. Messaoudi. Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021038

[9]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

[10]

Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130

[11]

Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361

[12]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079

[13]

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

[14]

Lorenzo Brasco, Marco Squassina, Yang Yang. Global compactness results for nonlocal problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 391-424. doi: 10.3934/dcdss.2018022

[15]

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

[16]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143

[17]

Lele Du, Minbo Yang. Uniqueness and nondegeneracy of solutions for a critical nonlocal equation. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5847-5866. doi: 10.3934/dcds.2019219

[18]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058

[19]

Xiyou Cheng, Zhaosheng Feng, Lei Wei. Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3067-3083. doi: 10.3934/dcdss.2021078

[20]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (202)
  • HTML views (241)
  • Cited by (0)

Other articles
by authors

[Back to Top]