A critical Kirchhoff problem with a logarithmic type perturbation in high dimension

Qi Li; Yuzhu Han; Bin Guo; Qi Li; Yuzhu Han; Bin Guo

doi:10.3934/cam.2024027

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 578-598. doi: 10.3934/cam.2024027

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A critical Kirchhoff problem with a logarithmic type perturbation in high dimension

Qi Li ¹,
Yuzhu Han ²,
Bin Guo ^{2
,
,}

1.
Department of Mathematics, Dalian Minzu University, Dalian 116600, P.R. China
2.
School of Mathematics, Jilin University, Changchun 130012, P.R. China

Received: 09 November 2023 Revised: 22 January 2024 Accepted: 05 August 2024 Published: 19 August 2024
35J60, 35B33

In this paper, the following critical Kirchhoff-type elliptic equation involving a logarithmic-type perturbation
$ -\Big(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x\Big)\Delta u = \lambda|u|^{q-2}u\ln |u|^2+\mu|u|^2u $
is considered in a bounded domain in $ \mathbb{R}^{4} $. One of the main obstructions one encounters when looking for weak solutions to Kirchhoff problems in high dimensions is that the boundedness of the $ (PS) $ sequence is hard to obtain. By combining a result by Jeanjean ^[27] with the mountain pass lemma and Brézis–Lieb's lemma, it is proved that either the norm of the sequence of approximation solutions goes to infinity or the problem admits a nontrivial weak solution, under some appropriate assumptions on $ a $, $ b $, $ \lambda $, and $ \mu $.
- critical,
- Kirchhoff equation,
- logarithmic perturbation,
- high dimension,
- mountain pass lemma
Citation: Qi Li, Yuzhu Han, Bin Guo. A critical Kirchhoff problem with a logarithmic type perturbation in high dimension[J]. Communications in Analysis and Mechanics, 2024, 16(3): 578-598. doi: 10.3934/cam.2024027

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Abstract

In this paper, the following critical Kirchhoff-type elliptic equation involving a logarithmic-type perturbation

$ -\Big(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x\Big)\Delta u = \lambda|u|^{q-2}u\ln |u|^2+\mu|u|^2u $

is considered in a bounded domain in $ \mathbb{R}^{4} $. One of the main obstructions one encounters when looking for weak solutions to Kirchhoff problems in high dimensions is that the boundedness of the $ (PS) $ sequence is hard to obtain. By combining a result by Jeanjean ^[27] with the mountain pass lemma and Brézis–Lieb's lemma, it is proved that either the norm of the sequence of approximation solutions goes to infinity or the problem admits a nontrivial weak solution, under some appropriate assumptions on $ a $, $ b $, $ \lambda $, and $ \mu $.

References

[1]	G. Kirchhoff, Mechanik Teubner, Leipzig, 1883.
[2]	A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305–330.
[3]	M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
[4]	A. T. Cousin, C. L. Frota, N. A. Lar'kin, L. A. Mediros, On the abstract model of the Kirchhoff-Carrier equation, Commun. Appl. Anal., 1 (1997), 389–404.
[5]	X. Ke, J. Liao, J. Liu, Positive solutions for a critical $p$-Laplacian problem with a Kirchhoff term, Comput. Math. Appl., 77(2019), 2279–2290. https://doi.org/10.1016/j.camwa.2018.12.021 doi: 10.1016/j.camwa.2018.12.021
[6]	K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 221 (2006), 246–255. https://doi.org/10.1016/j.jde.2005.03.006 doi: 10.1016/j.jde.2005.03.006
[7]	W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equ., 259 (2015), 1256–1274. https://doi.org/10.1016/j.jde.2015.02.040 doi: 10.1016/j.jde.2015.02.040
[8]	X. Tang, B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differ. Equ., 261 (2016), 2384–2402. https://doi.org/10.1016/j.jde.2016.04.032 doi: 10.1016/j.jde.2016.04.032
[9]	Q. Xie, S. Ma, X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differ. Equ., 261 (2016), 890–924. https://doi.org/10.1016/j.jde.2016.03.028 doi: 10.1016/j.jde.2016.03.028
[10]	F. Zhang, M. Du, Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equ., 269 (2020), 10085–10106. https://doi.org/10.1016/j.jde.2020.07.013 doi: 10.1016/j.jde.2020.07.013
[11]	C. O. Alves, F. J. S. A. Correa, T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85–93. https://doi.org/10.1016/j.camwa.2005.01.008 doi: 10.1016/j.camwa.2005.01.008
[12]	C. Chen, Y. Kuo, T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250 (2011), 1876–1908. https://doi.org/10.1016/j.jde.2010.11.017 doi: 10.1016/j.jde.2010.11.017
[13]	K. Silva, The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material, Z. Angew. Math. Phys., 70 (2019), 70–93. https://doi.org/10.1007/s00033-019-1137-8 doi: 10.1007/s00033-019-1137-8
[14]	Q. Li, Y. Han, Existence and multiplicity of positive solutions to a $p$-Kirchhoff-type equation, Bull. Malays. Math. Sci. Soc., 45 (2022), 1789–1810. https://doi.org/10.1007/s40840-022-01278-0 doi: 10.1007/s40840-022-01278-0
[15]	F. Faraci, C. Farkas, On an open question of Ricceri concerning a Kirchhoff-type problem, Minimax Theory Appl., 4 (2019), 271–280. https://doi.org/10.48550/arXiv.1810.08224 doi: 10.48550/arXiv.1810.08224
[16]	D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differ. Equ., 257 (2014), 1168–1193. https://doi.org/10.1016/j.jde.2014.05.002 doi: 10.1016/j.jde.2014.05.002
[17]	D. Naimen, M. Shibata, Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity in high dimension, Nonlinear Anal., 186 (2019), 187–208. https://doi.org/10.1016/j.na.2019.02.003 doi: 10.1016/j.na.2019.02.003
[18]	F. Faraci, K. Silva, On the Brezis-Nirenberg problem for a Kirchhoff type equation in high dimension, Calc. Var. Partial Differential Equations, 60 (2021), 1–33. https://doi.org/10.1007/s00526-020-01891-6 doi: 10.1007/s00526-020-01891-6
[19]	Y. Li, F. Li, J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285–2294. https://doi.org/10.1016/j.jde.2012.05.017 doi: 10.1016/j.jde.2012.05.017
[20]	Y. Chen, R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664. https://doi.org/10.1016/j.na.2019.111664 doi: 10.1016/j.na.2019.111664
[21]	W. Lian, M. Ahmed, R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239–257. https://doi.org/10.1016/j.na.2019.02.015 doi: 10.1016/j.na.2019.02.015
[22]	X. Wang, Y. Chen, Y. Yang, J. Li, R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475–499. https://doi.org/10.1016/j.na.2019.06.019 doi: 10.1016/j.na.2019.06.019
[23]	H. Yang, Y. Han, Blow-up for a damped $p$-Laplacian type wave equation with logarithmic nonlinearity, J. Differ. Equ., 306 (2022), 569–589. https://doi.org/10.1016/j.jde.2021.10.036 doi: 10.1016/j.jde.2021.10.036
[24]	W. Shuai, Two sequences of solutions for the semilinear elliptic equations with logarithmic nonlinearities, J. Differ. Equ., 343 (2023), 263–284. https://doi.org/10.1016/j.jde.2022.10.014 doi: 10.1016/j.jde.2022.10.014
[25]	Q. Li, Y. Han, T. Wang, Existence and nonexistence of solutions to a critical biharmonic equation with logarithmic perturbation, J. Differ. Equ., 365 (2023), 1–37. https://doi.org/10.1016/j.jde.2023.04.003 doi: 10.1016/j.jde.2023.04.003
[26]	Q. Zhang, Y. Han, J. Wang, A note on a critical bi-harmonic equation with logarithmic perturbation, Appl. Math. Lett., 145 (2023), 108784. https://doi.org/10.1016/j.aml.2023.108784 doi: 10.1016/j.aml.2023.108784
[27]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787–809. https://doi.org/10.1017/S0308210500013147 doi: 10.1017/S0308210500013147
[28]	M. Willem, Minimax theorems, Birkhäuser, Boston, 1996.
[29]	H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.
[30]	H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437–477. https://doi.org/10.1002/cpa.3160360405 doi: 10.1002/cpa.3160360405
[31]	S. I. Pohozaev, Eigenfunctions of the equation $\Delta u+\lambda f(u) = 0$, Soviet Math. Dokl., 6(1965), 1408–1411.

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