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On the compactness threshold in the critical Kirchhoff equation
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA |
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.
References:
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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. |
[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. |
[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. |
[4] |
H. Brezis and L. Nirenberg,
Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.
doi: 10.1002/cpa.3160440808. |
[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. |
[6] |
E. Hebey,
Compactness and the Palais-Smale property for critical Kirchhoff equations in closed manifolds, Pacific J. Math., 280 (2016), 41-50.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
H. Brezis and L. Nirenberg,
Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.
doi: 10.1002/cpa.3160440808. |
[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. |
[6] |
E. Hebey,
Compactness and the Palais-Smale property for critical Kirchhoff equations in closed manifolds, Pacific J. Math., 280 (2016), 41-50.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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