Existence and multiplicity results for a singular fourth-order elliptic system involving critical homogeneous nonlinearities

Zhiying Deng; Yisheng Huang; Zhiying Deng; Yisheng Huang

doi:10.3934/math.2023453

AIMS Mathematics

2023, Volume 8, Issue 4: 9054-9073. doi: 10.3934/math.2023453

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Existence and multiplicity results for a singular fourth-order elliptic system involving critical homogeneous nonlinearities

Zhiying Deng ^{1
,
,},
Yisheng Huang ²

1.
School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China

Received: 28 October 2022 Revised: 02 February 2023 Accepted: 05 February 2023 Published: 10 February 2023
MSC : 35J35, 35J40, 35J50

This paper deals with a singular fourth-order elliptic system involving critical homogeneous nonlinearities. The existence and multiplicity results of group invariant solutions are established by variational methods and the Hardy-Rellich inequality.
- Hardy-Rellich inequality,
- group invariant solution,
- variational methods,
- fourth-order elliptic system
Citation: Zhiying Deng, Yisheng Huang. Existence and multiplicity results for a singular fourth-order elliptic system involving critical homogeneous nonlinearities[J]. AIMS Mathematics, 2023, 8(4): 9054-9073. doi: 10.3934/math.2023453

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Abstract

This paper deals with a singular fourth-order elliptic system involving critical homogeneous nonlinearities. The existence and multiplicity results of group invariant solutions are established by variational methods and the Hardy-Rellich inequality.

References

[1]	W. D. Bastos, O. H. Miyagaki, R. S. Vieira, Solution to biharmonic equation with vanishing potential, Illinois J. Math., 57 (2013), 839–854. https://doi.org/10.1215/IJM/1415023513 doi: 10.1215/IJM/1415023513
[2]	M. Badiale, S. Greco, S. Rolando, Radial solutions of a biharmonic equation with vanishing or singular radial potentials, Nonlinear Anal., 185 (2019), 97–122. https://doi.org/10.1016/j.na.2019.01.011 doi: 10.1016/j.na.2019.01.011
[3]	O. H. Miyagaki, C. R. Santana, R. S. Vieira, Schrödinger equations in $\mathbb{R}^{4}$ involving the biharmonic operator with critical exponential growth, Rocky Mountain J. Math., 51 (2021), 243–263. https://doi.org/10.1216/rmj.2021.51.243 doi: 10.1216/rmj.2021.51.243
[4]	A. Rani, S. Goyal, Polyharmonic systems involving critical nonlinearities with sign-changing weight functions, Electron. J. Differ. Eq., 2020 (2020), 119.
[5]	H. S. Zhang, T. Li, T. Wu, Existence and multiplicity of nontrivial solutions for biharmonic equations with singular weight functions, Appl. Math. Lett., 105 (2020), 106335. https://doi.org/10.1016/j.aml.2020.106335 doi: 10.1016/j.aml.2020.106335
[6]	Y. Su, H. Shi, Ground state solution of critical biharmonic equation with Hardy potential and $p$-Laplacian, Appl. Math. Lett., 112 (2021), 106802. https://doi.org/10.1016/j.aml.2020.106802 doi: 10.1016/j.aml.2020.106802
[7]	Z. Feng, Y. Su, Ground state solution to the biharmonic equation, Z. Angew. Math. Phys., 73 (2022), 15. https://doi.org/10.1007/s00033-021-01643-2 doi: 10.1007/s00033-021-01643-2
[8]	Y. Yu, Y. Zhao, C. Luo, Ground state solution of critical $p$-biharmonic equation involving Hardy potential, Bull. Malays. Math. Sci. Soc., 45 (2022), 501–512. https://doi.org/10.1007/s40840-021-01192-x doi: 10.1007/s40840-021-01192-x
[9]	Y. X. Yao, R. S. Wang, Y. T. Shen, Nontrivial solutions for a class of semilinear biharmonic equation involving critical exponents, Acta Math. Sci., 27 (2007), 509–514. https://doi.org/10.1016/S0252-9602(07)60050-2 doi: 10.1016/S0252-9602(07)60050-2
[10]	L. D'Ambrosio, E. Jannelli, Nonlinear critical problems for the biharmonic operator with Hardy potential, Calc. Var. Partial Dif., 54 (2015), 365–396. https://doi.org/10.1007/s00526-014-0789-7 doi: 10.1007/s00526-014-0789-7
[11]	Z. Y. Deng, Y. S. Huang, Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents, Appl. Math. Comput., 264 (2015), 323–334. https://doi.org/10.1016/j.amc.2015.04.099 doi: 10.1016/j.amc.2015.04.099
[12]	D. Kang, L. Xu, Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities, Commun. Pur. Appl. Anal., 17 (2018), 333–346. https://doi.org/10.3934/cpaa.2018019 doi: 10.3934/cpaa.2018019
[13]	D. C. de Morais Filho, M. A. S. Souto, Systems of $p$-Laplacean equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Commun. Part. Diff. Eq., 24 (1999), 1537–1553. https://doi.org/10.1080/03605309908821473 doi: 10.1080/03605309908821473
[14]	H. Brezis, L. Nirenberg, Postive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pur. Appl. Math., 36 (1983), 437–477. https://doi.org/10.1002/cpa.3160360405 doi: 10.1002/cpa.3160360405
[15]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1, Rev. Mat. Iberoam., 1 (1985), 145–201. https://doi.org/10.4171/RMI/6 doi: 10.4171/RMI/6
[16]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 2, Rev. Mat. Iberoam., 1 (1985), 45–121. https://doi.org/10.4171/RMI/12 doi: 10.4171/RMI/12
[17]	M. F. Furtado, J. P. P. da Silva, Multiplicity of solutions for homogeneous elliptic systems with critical growth, J. Math. Anal. Appl., 385 (2012), 770–785. https://doi.org/10.1016/j.jmaa.2011.07.001 doi: 10.1016/j.jmaa.2011.07.001
[18]	J. P. P. da Silva, C. P. de Oliveira, Existence and multiplicity of positive solutions for a critical weight elliptic system in $\mathbb{R}^{N}$, Commun. Contemp. Math., 23 (2021), 2050027. https://doi.org/10.1142/S0219199720500273 doi: 10.1142/S0219199720500273
[19]	G. M. Figueiredo, S. M. A. Salirrosas, On multiplicity and concentration behavior of solutions for a critical system with equations in divergence form, J. Math. Anal. Appl., 494 (2021), 124446. https://doi.org/10.1016/j.jmaa.2020.124446 doi: 10.1016/j.jmaa.2020.124446
[20]	M. F. Furtado, L. D. de Oliveira, J. P. P. da Silva, Existence and multiplicity of solutions for a Kirchhoff system with critical growth, Ann. Fenn. Math., 46 (2021), 295–308.
[21]	V. P. Bandeira, G. M. Figueiredo, On a critical and a supercritical system with fast increasing weights, Nonlinear Anal. Real, 64 (2022), 103431. https://doi.org/10.1016/j.nonrwa.2021.103431 doi: 10.1016/j.nonrwa.2021.103431
[22]	D. S. Kang, P. Xiong, Existence and nonexistence results for critical biharmonic systems involving multiple singularities, J. Math. Anal. Appl., 452 (2017), 469–487. https://doi.org/10.1016/j.jmaa.2017.03.011 doi: 10.1016/j.jmaa.2017.03.011
[23]	T. Zheng, P. Ma, J. Zhang, Non-existence of positive solutions for a class of fourth order elliptic systems in positive-type domains, Appl. Math. Lett., 117 (2021), 107085. https://doi.org/10.1016/j.aml.2021.107085 doi: 10.1016/j.aml.2021.107085
[24]	T. Yang, On a critical biharmonic system involving $p$-Laplacian and Hardy potential, Appl. Math. Lett., 121 (2021), 107433. https://doi.org/10.1016/j.aml.2021.107433 doi: 10.1016/j.aml.2021.107433
[25]	G. Bianchi, J. Chabrowski, A. Szulkin, On symmetric solutions of an elliptic equations with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. Theor., 25 (1995), 41–59. https://doi.org/10.1016/0362-546X(94)E0070-W doi: 10.1016/0362-546X(94)E0070-W
[26]	Z. Y. Deng, R. Zhang, Y. S. Huang, Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity, Math. Method. Appl. Sci., 40 (2017), 1538–1552. https://doi.org/10.1002/mma.4078 doi: 10.1002/mma.4078
[27]	Y. Wang, Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027–1043. https://doi.org/10.1016/j.jmaa.2017.10.015 doi: 10.1016/j.jmaa.2017.10.015
[28]	L. Baldelli, R. Filippucci, Singular quasilinear critical Schrödinger equations in $\mathbb{R}^{N}$, Commun. Pur. Appl. Anal., 21 (2022), 2561–2586. https://doi.org/10.3934/cpaa.2022060 doi: 10.3934/cpaa.2022060
[29]	L. Baldelli, Y. Brizi, R. Filippucci, On symmetric solutions for $(p, q)$-Laplacian equations in $\mathbb{R}^{N}$ with critical terms, J. Geom. Anal., 32 (2022), 120. https://doi.org/10.1007/s12220-021-00846-3 doi: 10.1007/s12220-021-00846-3
[30]	L. D'Ambrosio, E. Jannelli, Nonlinear critical problems for the biharmonic operator with Hardy potential, Calc. Var. Partial Dif., 54 (2015), 365–396. https://doi.org/10.1007/s00526-014-0789-7 doi: 10.1007/s00526-014-0789-7
[31]	D. S. Kang, P. Xiong, Ground state solutions to biharmonic equations involving critical nonlinearities and multiple singular potentials, Appl. Math. Lett., 66 (2017), 9–15. https://doi.org/10.1016/j.aml.2016.10.014 doi: 10.1016/j.aml.2016.10.014
[32]	M. Willem, Minimax theorems, Boston: Birkhäuser, 1996. https://doi.org/10.1007/978-1-4612-4146-1
[33]	A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
[34]	H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathmatical Society, 1986.
[35]	H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490. https://doi.org/10.1090/S0002-9939-1983-0699419-3 doi: 10.1090/S0002-9939-1983-0699419-3

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