Multiplicity and concentration of normalized solutions for a Kirchhoff type problem with $ L^2 $-subcritical nonlinearities

Yangyu Ni; Jijiang Sun; Jianhua Chen; Yangyu Ni; Jijiang Sun; Jianhua Chen

doi:10.3934/cam.2024029

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 633-654. doi: 10.3934/cam.2024029

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Multiplicity and concentration of normalized solutions for a Kirchhoff type problem with $ L^2 $-subcritical nonlinearities

Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, P. R. China

Received: 06 January 2024 Revised: 18 June 2024 Accepted: 12 August 2024 Published: 09 September 2024
35J50, 35J93, 35Q60

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation:
$ \begin{equation*} \begin{cases} -\left(a\varepsilon^2+b\varepsilon\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+V(x)u = \mu u+f(u) & {\rm{in}}\;\mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx = m\varepsilon^3 , u\in H^1(\mathbb{R}^3) , \end{cases} \end{equation*} $
where $ a $, $ b $, $ m > 0 $, $ \varepsilon $ is a small positive parameter, $ V $ is a nonnegative continuous function, $ f $ is a continuous function with $ L^2 $-subcritical growth and $ \mu\in\mathbb{R} $ will arise as a Lagrange multiplier. Under the suitable assumptions on $ V $ and $ f $, the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $ V $ attained its minimum value.
- Kirchhoff type equation,
- normalized solutions,
- multiplicity,
- mass subcritical,
- Lusternik- Schnirelman category
Citation: Yangyu Ni, Jijiang Sun, Jianhua Chen. Multiplicity and concentration of normalized solutions for a Kirchhoff type problem with $ L^2 $-subcritical nonlinearities[J]. Communications in Analysis and Mechanics, 2024, 16(3): 633-654. doi: 10.3934/cam.2024029

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Abstract

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation:

$ \begin{equation*} \begin{cases} -\left(a\varepsilon^2+b\varepsilon\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+V(x)u = \mu u+f(u) & {\rm{in}}\;\mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx = m\varepsilon^3 , u\in H^1(\mathbb{R}^3) , \end{cases} \end{equation*} $

where $ a $, $ b $, $ m > 0 $, $ \varepsilon $ is a small positive parameter, $ V $ is a nonnegative continuous function, $ f $ is a continuous function with $ L^2 $-subcritical growth and $ \mu\in\mathbb{R} $ will arise as a Lagrange multiplier. Under the suitable assumptions on $ V $ and $ f $, the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $ V $ attained its minimum value.

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]	S. Bernstein, Sur une classe d'$\acute{e}$quations fonctionelles aux d$\acute{e}$riv$\acute{e}$es partielles, Izv. Akad. Nauk SSSR Ser. Mat, 4 (1940), 17–26.
[4]	S. I. Poho$\check{z}$aev, On a class of quasilinear hyperbolic equations, sb. Math., 25 (1975), 145–158.
[5]	G. Li, H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differ. Equ., 257 (2014), 566–600. https://doi.org/10.1016/j.jde.2014.04.011 doi: 10.1016/j.jde.2014.04.011
[6]	G. Li, P. Luo, S. Peng, C. Wang, C. L. Xiang, A singularly perturbed Kirchhoff problem revisited, J. Differ. Equ., 268 (2020), 541–589. https://doi.org/10.1016/j.jde.2019.08.016 doi: 10.1016/j.jde.2019.08.016
[7]	J. Wang, L. Tian, J. Xu, F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314–2351. https://doi.org/10.1016/j.jde.2012.05.023 doi: 10.1016/j.jde.2012.05.023
[8]	X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 252 (2012), 1813–1834. https://doi.org/10.1016/J.JDE.2011.08.035 doi: 10.1016/J.JDE.2011.08.035
[9]	T. Hu, W. Shuai, Multi-peak solutions to Kirchhoff equations in $\mathbb{R}^3$ with general nonlinearity, J. Differ. Equ., 265 (2018), 3587–3617. https://doi.org/10.1016/j.jde.2018.05.012 doi: 10.1016/j.jde.2018.05.012
[10]	G. M. Figueiredo, N. Ikoma, J. R. Santos J$\acute{u}$nior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931–979. https://doi.org/10.1007/s00205-014-0747-8 doi: 10.1007/s00205-014-0747-8
[11]	Y. He, G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var., 54 (2015), 3067–3106. https://doi.org/10.1007/s00526-015-0894-2 doi: 10.1007/s00526-015-0894-2
[12]	Q. Xie, X. Zhang, Semi-classical solutions for Kirchhoff type problem with a critical frequency, Proc. Roy. Soc. Edinburgh Sect. A., 151 (2021), 761–798. https://doi.org/10.1017/prm.2020.37 doi: 10.1017/prm.2020.37
[13]	L. Kong, H. Chen, Normalized ground states for fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities, J. Math. Phys. 64 (2023), 061501. https://doi.org/10.1063/5.0098126 doi: 10.1063/5.0098126
[14]	L. Kong, L. Zhu, Y. Deng, Normalized solutions for nonlinear Kirchhoff type equations with low-order fractional Laplacian and critical exponent, Appl. Math. Lett., 147 (2023), 108864. https://doi.org/10.1016/j.aml.2023.108864 doi: 10.1016/j.aml.2023.108864
[15]	S. Chen, V. Rădulescu, X. Tang, Normalized solutions of nonautonomous Kirchhoff equations: sub- and super-critical cases, Appl. Math. Optim., 84 (2021), 773–806. https://doi.org/10.1007/s00245-020-09661-8 doi: 10.1007/s00245-020-09661-8
[16]	J. Hu, J. Sun, Normalized ground states for Kirchhoff type equations with general nonlinearities, Adv. Differential Equ., 29 (2024), 111–152. https://doi.org/10.57262/ade029-0102-111 doi: 10.57262/ade029-0102-111
[17]	T. Hu, C. L. Tang, Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations, Calc. Var., 60 (2021), 210. https://doi.org/10.1007/s00526-021-02018-1 doi: 10.1007/s00526-021-02018-1
[18]	Q. Li, J. Nie, W. Zhang, Existence and asymptotics of normalized ground states for a Sobolev critical Kirchhoff equation, J. Geom. Anal., 33 (2023), 126. https://doi.org/10.1007/s12220-022-01171-z doi: 10.1007/s12220-022-01171-z
[19]	Q. Li, V. D. Radulescu, W. Zhang, Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth, Nonlinearity, 37 (2024), 025018. https://doi.org/10.1088/1361-6544/ad1b8b doi: 10.1088/1361-6544/ad1b8b
[20]	G. Li, H. Ye, On the concentration phenomenon of $L^2$-subcritical constrained minimizers for a class of Kirchhoff equations with potentials, J. Differ. Equ., 266 (2019), 7101–7123. https://doi.org/10.1016/j.jde.2018.11.024 doi: 10.1016/j.jde.2018.11.024
[21]	S. Qi, W. Zou, Exact Number of Positive Solutions for the Kirchhoff Equation, SIAM J. Math. Anal., 54 (2022), 5424–5446. https://doi.org/10.1137/21M1445879 doi: 10.1137/21M1445879
[22]	H. Ye, The existence of normalized solutions for $L^2$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 66 (2015), 1483–1497. https://doi.org/10.1007/s00033-014-0474-x doi: 10.1007/s00033-014-0474-x
[23]	H. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci., 38 (2015), 2663–2679. https://doi.org/10.1002/mma.3247 doi: 10.1002/mma.3247
[24]	H. Ye, The mass concentration phenomenon for $L^2$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phy., 67 (2016), 29. https://doi.org/10.1007/s00033-016-0624-4 doi: 10.1007/s00033-016-0624-4
[25]	X. Zeng, J. Zhang, Y. Zhang, X. Zhong, On the Kirchhoff equation with prescribed mass and general nonlinearities, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 3394–3409. https://doi.org/10.3934/dcdss.2023160 doi: 10.3934/dcdss.2023160
[26]	X. Zeng, Y. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett., 74 (2017), 52–59. https://doi.org/10.1016/j.aml.2017.05.012 doi: 10.1016/j.aml.2017.05.012
[27]	C. O. Alves, N. V. Thin, On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^{N}$ via Lusternik-Schnirelmann Category, SIAM J. Math. Anal., 55 (2023), 1264–1283. https://doi.org/10.1137/22M1470694 doi: 10.1137/22M1470694
[28]	C. O. Alves, N. V. Thin, On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^{N}$ via penalization method, Potential Anal., 2023. https://doi.org/10.1007/s11118-023-10116-2 doi: 10.1007/s11118-023-10116-2
[29]	N. Ackermann, T. Weth, Unstable normalized standing waves for the space periodic NLS, Anal. PDE., 12 (2018), 1177–1213. https://doi.org/10.2140/apde.2019.12.1177 doi: 10.2140/apde.2019.12.1177
[30]	C. O. Alves, On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^N$, Z. Angew. Math. Phys., 73 (2022), 97. https://doi.org/10.1007/s00033-022-01741-9 doi: 10.1007/s00033-022-01741-9
[31]	B. Pellacci, A. Pistoia, G. Vaira, G. Verzini, Normalized concentrating solutions to nonlinear elliptic problems, J. Differ. Equ., 275 (2021), 882–919. https://doi.org/10.1016/j.jde.2020.11.003 doi: 10.1016/j.jde.2020.11.003
[32]	N. S. Papageorgiou, J. Zhang, W. Zhang, Solutions with sign information for noncoercive double phase equations, J. Geom. Anal., 34 (2024), 14. https://doi.org/10.1007/s12220-023-01463-y doi: 10.1007/s12220-023-01463-y
[33]	Z. Tang, C. Zhang, L. Zhang, L. Zhou, Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well, Nonlinearity, 35 (2022), 4624. https://doi.org/10.1088/1361-6544/ac7b61 doi: 10.1088/1361-6544/ac7b61
[34]	C. Zhang, X. Zhang, Normalized multi-bump solutions of nonlinear Schrödinger equations via variational approach, Calc. Var., 61 (2022), 57. https://doi.org/10.1007/s00526-021-02166-4 doi: 10.1007/s00526-021-02166-4
[35]	J. Zhang, W. Zhang, Semiclassical states for coupled nonlinear Schrödinger system with competing potentials, J. Geom. Anal., 32 (2022), 114. https://doi.org/10.1007/s12220-022-00870-x doi: 10.1007/s12220-022-00870-x
[36]	J. Hu, J. Sun, On constrained minimizers for Kirchhoff type equations with Berestycki-Lions type mass subcritical conditions, Electron. Res. Arch., 31 (2023), 2580–2594. https://doi.org/10.3934/era.2023131 doi: 10.3934/era.2023131
[37]	M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1
[38]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commu. Math. Phys., 87 (1983), 567–576. https://doi.org/10.1007/BF01208265 doi: 10.1007/BF01208265
[39]	H. Berestycki, P. L. Lions, Nonlinear scalar field equations Ⅰ: Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313–346. https://doi.org/10.1007/BF00250555 doi: 10.1007/BF00250555
[40]	M. Shibata, Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscripta Math., 143 (2014), 221–237. https://doi.org/10.1007/s00229-013-0627-9 doi: 10.1007/s00229-013-0627-9
[41]	D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, springer, Berlin, 1977.
[42]	V. Benci, G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var., 2 (1994), 29–48. https://doi.org/10.1007/BF01234314 doi: 10.1007/BF01234314
[43]	N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge University Press, 1993.

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