This paper dealt with a class of Kirchhoff type equations involving singular nonlinearity in a closed Riemannian manifold $ (M, g) $ of dimension $ n\ge3 $. Existence and uniqueness of a positive weak solution were obtained under certain assumptions with the help of the variation methods and some analysis techniques.
Citation: Nanbo Chen, Honghong Liang, Xiaochun Liu. On Kirchhoff type problems with singular nonlinearity in closed manifolds[J]. AIMS Mathematics, 2024, 9(8): 21397-21413. doi: 10.3934/math.20241039
This paper dealt with a class of Kirchhoff type equations involving singular nonlinearity in a closed Riemannian manifold $ (M, g) $ of dimension $ n\ge3 $. Existence and uniqueness of a positive weak solution were obtained under certain assumptions with the help of the variation methods and some analysis techniques.
[1] | G. Kirchhoff, Vorlesungen über Mathematische Physik, 1883. |
[2] |
J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284–346. http://dx.doi.org/10.1016/S0304-0208(08)70870-3 doi: 10.1016/S0304-0208(08)70870-3
![]() |
[3] | A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305–330. |
[4] |
C. O. Alves, F. J. S. A. Corrêa, G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 3 (2010), 409–417. http://dx.doi.org/10.7153/dea-02-25 doi: 10.7153/dea-02-25
![]() |
[5] |
H. N. Fang, X. C. Liu, On the multiplicity and concentration of positive solutions to a Kirchhoff-type problem with competing potentials, J. Math. Phys., 63 (2022), 011512. http://dx.doi.org/10.1063/5.0073716 doi: 10.1063/5.0073716
![]() |
[6] |
A. Fiscella, A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal., 8 (2019), 645–660. http://dx.doi.org/10.1515/anona-2017-0075 doi: 10.1515/anona-2017-0075
![]() |
[7] |
X. M. He, W. M. Zou, Infinitely many solutions for Kirchhoff type problems, Nonlinear Anal., 70 (2009), 1407–1414. http://dx.doi.org/10.1016/j.na.2008.02.021 doi: 10.1016/j.na.2008.02.021
![]() |
[8] |
Y. J. Sun, Y. X. Tan, Kirchhoff type equations with strong singularities, Commun. Pur. Appl. Anal., 18 (2019), 181–193. http://dx.doi.org/10.3934/cpaa.2019010 doi: 10.3934/cpaa.2019010
![]() |
[9] |
D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differ. Equ., 257 (2014), 1168–1193. http://dx.doi.org/10.1016/j.jde.2014.05.002 doi: 10.1016/j.jde.2014.05.002
![]() |
[10] |
D. Naimen, On the Brezis-Nirenberg problem with a Kirchhoff type perturbation, Adv. Nonlinear Stud., 15 (2015), 135–156. http://dx.doi.org/10.1515/ans-2015-0107 doi: 10.1515/ans-2015-0107
![]() |
[11] |
F. Faraci, K. Silva, On the Brezis-Nirenberg problem for a Kirchhoff type equation in high dimension, Calc. Var. Partial. Dif., 60 (2021), 1–33. http://dx.doi.org/10.1007/s00526-020-01891-6 doi: 10.1007/s00526-020-01891-6
![]() |
[12] |
X. Liu, Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pur. Appl. Anal., 12 (2013), 721–733. http://dx.doi.org/10.3934/cpaa.2013.12.721 doi: 10.3934/cpaa.2013.12.721
![]() |
[13] |
C. Y. Lei, J. F. Liao, C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521–538. http://doi.org/10.1016/j.jmaa.2014.07.031 doi: 10.1016/j.jmaa.2014.07.031
![]() |
[14] |
Y. Duan, H. Y. Li, X. Sun, Uniqueness of positive solutions for a class of $p$-Kirchhoff type problems with singularity, Rocky Mountain J. Math., 51 (2021), 1629–1637. http://dx.doi.org/10.1216/rmj.2021.51.1629 doi: 10.1216/rmj.2021.51.1629
![]() |
[15] |
D. C. Wang, B. Q. Yan, A uniqueness result for some Kirchhoff-type equations with negative exponents, Appl. Math. Lett., 92 (2019), 93–98. http://dx.doi.org/10.1016/j.aml.2019.01.002 doi: 10.1016/j.aml.2019.01.002
![]() |
[16] |
N. B. Chen, X. C. Liu. Hardy-Sobolev equation on compact Riemannian manifolds involving $p$-Laplacian, J. Math. Anal. Appl., 487 (2020), 123992. https://doi.org/10.1016/j.jmaa.2020.123992 doi: 10.1016/j.jmaa.2020.123992
![]() |
[17] |
N. B. Chen, X. C. Liu. On the $p$-Laplacian Lichnerowicz equation on compact Riemannian manifolds, Sci. China Math., 64 (2021), 2249–2274. https://doi.org/10.1007/s11425-020-1679-5 doi: 10.1007/s11425-020-1679-5
![]() |
[18] | A. Aberqi, O. Benslimane, A. Ouaziz, D. D. Repovs, On a new fractional Sobolev space with variable exponent on complete manifolds, Bound. Value Probl., 7 (2022). https://doi.org/10.1186/s13661-022-01590-5 |
[19] |
A. Aberqi, A. Ouaziz, Morse's theory and local linking for a fractional ($p_1$(x.,), $p_2$ (x.,)): Laplacian problems on compact manifolds, J. Pseudo-Differ. Oper. Appl., 14 (2023), 41. https://doi.org/10.1007/s11868-023-00535-5 doi: 10.1007/s11868-023-00535-5
![]() |
[20] |
E. Hebey, Compactness and the Palais-Smale property for critical Kirchhoff equations in closed manifolds, Pac. J. Math., 280 (2016), 41–50. http://dx.doi.org/10.2140/pjm.2016.280.41 doi: 10.2140/pjm.2016.280.41
![]() |
[21] |
E. Hebey, Multiplicity of solutions for critical Kirchhoff type equations, Commun. Part. Diff. Eq., 41 (2016), 913–924. http://dx.doi.org/10.1080/03605302.2016.1183213 doi: 10.1080/03605302.2016.1183213
![]() |
[22] |
E. Hebey, Stationary Kirchhoff equations with powers, Adv. Calc. Var., 11 (2018), 139–160. http://dx.doi.org/10.1515/acv-2016-0025 doi: 10.1515/acv-2016-0025
![]() |
[23] |
E. Hebey, P. D. Thizy, Stationary Kirchhoff systems in closed 3-dimensional manifolds, Calc. Var. Partial. Dif., 54 (2015), 2085–2114. http://dx.doi.org/10.1007/s00526-015-0858-6 doi: 10.1007/s00526-015-0858-6
![]() |
[24] |
E. Hebey, P. D. Thizy, Stationary Kirchhoff systems in closed high dimensional manifolds, Commun. Contemp. Math., 18 (2016), 1550028. http://dx.doi.org/10.1142/S0219199715500285 doi: 10.1142/S0219199715500285
![]() |
[25] |
X. J. Bai, N. B. Chen, X. C. Liu. A class of critical $p$-Kirchhoff type equations on closed manifolds, Discrete Cont. Dyn. S., 44 (2024), 1087–1105. http://dx.doi.org/10.3934/dcds.2023139 doi: 10.3934/dcds.2023139
![]() |
[26] |
J. F. Liao, X. F. Ke, C. Y. Lei, C. L. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24–30. http://dx.doi.org/10.1016/j.aml.2016.03.001 doi: 10.1016/j.aml.2016.03.001
![]() |
[27] |
M. A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinb. Sect. A, 122 (1992), 341–352. http://dx.doi.org/10.1017/S0308210500021144 doi: 10.1017/S0308210500021144
![]() |
[28] | L. Boccardo, L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Dif., 37 (2010), 363–380. |
[29] |
Y. J. Sun, S. P. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257–1284. http://dx.doi.org/10.1016/j.jfa.2010.11.018 doi: 10.1016/j.jfa.2010.11.018
![]() |
[30] | O. Lablée, Spectral theory in Riemannian geometry, EMS Textbooks in Mathematics, 2015. http://dx.doi.org/10.4171/151 |
[31] |
H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490. http://dx.doi.org/10.1090/S0002-9939-1983-0699419-3 doi: 10.1090/S0002-9939-1983-0699419-3
![]() |