Existence and asymptotic behavior of normalized solutions for the mass supercritical fractional Kirchhoff equations with general nonlinearities

Min Shu; Haibo Chen; Jie Yang; Min Shu; Haibo Chen; Jie Yang

doi:10.3934/math.2025023

AIMS Mathematics

2025, Volume 10, Issue 1: 499-533. doi: 10.3934/math.2025023

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Research article

Existence and asymptotic behavior of normalized solutions for the mass supercritical fractional Kirchhoff equations with general nonlinearities

1.
School of Mathematics and Computational Science, Huaihua University, Huaihua, Hunan 418008, China
2.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received: 25 October 2024 Revised: 31 December 2024 Accepted: 03 January 2025 Published: 10 January 2025
MSC : 35A15, 35J20

In this paper, we studied a fractional Kirchhoff equation with mass supercritical general nonlinearities. Under some suitable conditions, we obtained the existence of ground state normalized solutions for this equation. Moreover, we presented the asymptotic behavior of normalized solutions to the above equation as $ c\to 0^{+} $ and $ c\to +\infty $.
- Kirchhoff equation,
- fractional,
- normalized solution,
- asymptotic behavior
Citation: Min Shu, Haibo Chen, Jie Yang. Existence and asymptotic behavior of normalized solutions for the mass supercritical fractional Kirchhoff equations with general nonlinearities[J]. AIMS Mathematics, 2025, 10(1): 499-533. doi: 10.3934/math.2025023

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Abstract

In this paper, we studied a fractional Kirchhoff equation with mass supercritical general nonlinearities. Under some suitable conditions, we obtained the existence of ground state normalized solutions for this equation. Moreover, we presented the asymptotic behavior of normalized solutions to the above equation as $ c\to 0^{+} $ and $ c\to +\infty $.

References

[1]	X. Bao, Y. Lv, Z. Ou, Normalized bound state solutions of fractional Schrödinger equations with general potential, Complex Var. Elliptic, in press. https://doi.org/10.1080/17476933.2024.2338436
[2]	M. Du, L. X. Tian, J. Wang, F. B. Zhang, Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials, Proc. Roy. Soc. Edinb. A, 149 (2019), 617–653. https://doi.org/10.1017/prm.2018.41 doi: 10.1017/prm.2018.41
[3]	P. Felmer, A. Quaas, J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinb. A, 142 (2012), 1237–1262. https://doi.org/10.1017/S0308210511000746 doi: 10.1017/S0308210511000746
[4]	R. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pur. Appl. Math., 69 (2016), 1671–1726. https://doi.org/10.1002/cpa.21591 doi: 10.1002/cpa.21591
[5]	R. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407–3430. https://doi.org/10.1016/j.jfa.2008.05.015 doi: 10.1016/j.jfa.2008.05.015
[6]	M. Fall, T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205–2227. http://dx.doi.org/10.1016/j.jfa.2012.06.018 doi: 10.1016/j.jfa.2012.06.018
[7]	H. L. Guo, Y. M. Zhang, H. S. Zhou, Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential, Commun. Pur. Appl. Anal., 17 (2018), 1875–1897. https://doi.org/10.3934/cpaa.2018089 doi: 10.3934/cpaa.2018089
[8]	Q. He, Z. Lv, Y. Zhang, X. Zhong, Existence and blow up behavior of positive normalized solution to the Kirchhoff equation with general nonlinearities: mass super-critical case, J. Differ. Equations, 356 (2023), 375–406. https://doi.org/10.1016/j.jde.2023.01.039 doi: 10.1016/j.jde.2023.01.039
[9]	N. Ikoma, K. Tanaka, A note on deformation argument for $L^{2}$ normalized solutions of nonlinear Schrödinger equations and systems, Adv. Differential Equations, 24 (2019), 609–646. https://doi.org/10.57262/ade/1571731543 doi: 10.57262/ade/1571731543
[10]	L. Z. Kong, H. B. 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
[11]	L. Li, K. Teng, J. Yang, H. Chen, Properties of minimizers for the fractional Kirchhoff energy functional, J. Math. Phys., 64 (2023), 081504. https://doi.org/ 10.1063/5.0157267 doi: 10.1063/5.0157267
[12]	L. T. Liu, H. B. Chen, J. Yang, Normalized solutions to the fractional Kirchhoff equations with a perturbation, Appl. Anal., 102 (2023), 1229–1249. https://doi.org/10.1080/00036811.2021.1979222 doi: 10.1080/00036811.2021.1979222
[13]	E. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
[14]	T. Phan, Blow-up profile of Bose-Einstein condensate with singular potentials, J. Math. Phys., 58 (2017), 072301. https://doi.org/10.1063/1.4995393 doi: 10.1063/1.4995393
[15]	S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. https://doi.org/10.1063/1.4793990 doi: 10.1063/1.4793990
[16]	Y. Su, Z. Liu, Semiclassical states to nonlinear Choquard equation with critical growth, Isr. J. Math., 255 (2023), 729–762. https://doi.org/10.1007/s11856-023-2485-9 doi: 10.1007/s11856-023-2485-9
[17]	Y. Su, Z. Liu, Semi-classical states for the Choquard equations with doubly critical exponents: existence, multiplicity and concentration, Asymptotic Anal., 132 (2023), 451–493. https://doi.org/10.3233/ASY-221799 doi: 10.3233/ASY-221799
[18]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pur. Appl. Math., 60 (2007), 67–112. https://doi.org/10.1002/cpa.20153 doi: 10.1002/cpa.20153
[19]	K. M. Teng, R. P. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Method. Appl. Sci., 41 (2018), 8258–8293. https://doi.org/10.1002/mma.5289 doi: 10.1002/mma.5289
[20]	J. Yang, H. Chen, L. Liu, Limiting behaviors of constrained minimizers for the mass subcritical fractional NLS equations, Anal. Math. Phys., 14 (2024), 32. https://doi.org/10.1007/s13324-024-00899-x doi: 10.1007/s13324-024-00899-x
[21]	S. Yao, H. Chen, V. Rădulescu, J. Sun, Normalized solutions for lower critical Choquard equations with critical Sobolev perturbations, SIAM J. Math. Anal., 54 (2022), 3696–3723. https://doi.org/10.1137/21M1463136 doi: 10.1137/21M1463136
[22]	H. Y. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Method. Appl. Sci., 38 (2015), 2663–2679. https://doi.org/10.1002/mma.3247 doi: 10.1002/mma.3247
[23]	H. Y. 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
[24]	X. Y. Zeng, Y. M. 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

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