Asymptotic behavior of the unique solution for a fractional Kirchhoff problem with singularity

Shengbin Yu; Jianqing Chen; Shengbin Yu; Jianqing Chen

doi:10.3934/math.2021421

AIMS Mathematics

2021, Volume 6, Issue 7: 7187-7198. doi: 10.3934/math.2021421

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Asymptotic behavior of the unique solution for a fractional Kirchhoff problem with singularity

Shengbin Yu ^{1
,
,},
Jianqing Chen ²

1.
Department of Basic Teaching and Research, Yango University, Fuzhou, Fujian 350015, China
2.
College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Qishan Campus, Fuzhou, Fujian 350117, China

Received: 20 September 2020 Accepted: 25 April 2021 Published: 28 April 2021
MSC : 35A15, 35R11

In this paper, we consider the following fractional Kirchhoff problem with singularity

$ \left \{\begin{array}{lcl} \Big(1+ b\int_{\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|u(x)-u(y)|^2}{|x-y|^{3+2s}}\mathrm{d}x \mathrm{d}y \Big)(-\Delta)^s u+V(x)u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. $

where $ (-\Delta)^s $ is the fractional Laplacian with $ 0 < s < 1 $, $ b\ge0 $ is a constant and $ 0 < \gamma < 1 $. Under certain assumptions on $ V $ and $ f $, we show the existence and uniqueness of positive solution $ u_b $ by using variational method. We also give a convergence property of $ u_b $ as $ b\rightarrow0 $, where $ b $ is regarded as a positive parameter.
- fractional Kirchhoff problem,
- singularity,
- uniqueness,
- variational method,
- asymptotic behavior
Citation: Shengbin Yu, Jianqing Chen. Asymptotic behavior of the unique solution for a fractional Kirchhoff problem with singularity[J]. AIMS Mathematics, 2021, 6(7): 7187-7198. doi: 10.3934/math.2021421

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Abstract

In this paper, we consider the following fractional Kirchhoff problem with singularity

$ \left \{\begin{array}{lcl} \Big(1+ b\int_{\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|u(x)-u(y)|^2}{|x-y|^{3+2s}}\mathrm{d}x \mathrm{d}y \Big)(-\Delta)^s u+V(x)u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. $

where $ (-\Delta)^s $ is the fractional Laplacian with $ 0 < s < 1 $, $ b\ge0 $ is a constant and $ 0 < \gamma < 1 $. Under certain assumptions on $ V $ and $ f $, we show the existence and uniqueness of positive solution $ u_b $ by using variational method. We also give a convergence property of $ u_b $ as $ b\rightarrow0 $, where $ b $ is regarded as a positive parameter.

References

[1]	V. Ambrosio, T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), 615–645.
[2]	D. Applebaum, L$\acute{e}$vy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336–1347.
[3]	G. Autuori, A. Fiscella, P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699–714. doi: 10.1016/j.na.2015.06.014
[4]	D. Barilla, G. Caristi, Existence results for some anisotropic Dirichlet problems, J. Math. Anal. Appl., 2020. Available from: https://doi.org/10.1016/j.jmaa.2020.124044.
[5]	G. Bisci, F. Tulone, An existence result for fractional Kirchhoff-type equations, Z. Anal. Anwendungen, 35 (2016), 181–197. doi: 10.4171/ZAA/1561
[6]	W. Chen, Y. Gui, Multiple solutions for a fractional $p$-Kirchhoff problem with Hardy nonlinearity, Nonlinear Anal., 188 (2019), 316–338. doi: 10.1016/j.na.2019.06.009
[7]	K. Cheng, Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $R^N$, Acta Math. Sci. Ser. B Engl. Ed., 38B (2018), 1712–1730.
[8]	G. Figueiredo, B. Molica, R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347–361. doi: 10.3233/ASY-151316
[9]	A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156–170. doi: 10.1016/j.na.2013.08.011
[10]	A. Fiscella, P. Pucci, Kirchhoff-Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud., 17 (2017), 429–456.
[11]	A. Fiscella, P. Mishra, The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms, Nonlinear Anal., 186 (2019), 6–32. doi: 10.1016/j.na.2018.09.006
[12]	A. Fiscella, A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal., 8 (2019), 645–660.
[13]	X. He, W. Zou, Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation, Manuscripta Math., 158 (2018), 159–203.
[14]	G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig, 1883.
[15]	C. Lei, J. Liao, C. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521–538. doi: 10.1016/j.jmaa.2014.07.031
[16]	C. Lei, J. Liao, Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities, Appl. Math. Lett., 94 (2019), 279–285. doi: 10.1016/j.aml.2019.03.007
[17]	F. Li, Z. Song, Q. Zhang, Existence and uniqueness results for Kirchhoff-Schrödinger-Poisson system with general singularity, Appl. Anal., 96 (2017), 2906–2916. doi: 10.1080/00036811.2016.1253065
[18]	H. Li, Y. Tang, J. Liao, Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign-changing potential, Math. Methods Appl. Sci., 41 (2018), 2971–2986. doi: 10.1002/mma.4795
[19]	Q. Li, W. Gao, Y. Han, Existence of solution for a singular elliptic equation of Kirchhoff type, Mediterr. J. Math., 14 (2017), 231. doi: 10.1007/s00009-017-1033-4
[20]	W. Li, V. Rǎdulescu, B. Zhang, Infinitely many solutions for fractional Kirchhoff-Schrödinger-Poisson systems, J. Math. Phys., 60 (2019), 011506. doi: 10.1063/1.5019677
[21]	J. Liao, P. Zhang, J. Liu, C. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124–1148. doi: 10.1016/j.jmaa.2015.05.038
[22]	J. Liao, X. Ke, C. Lei, C. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl. Math. Lett., 59 (2016), 24–30. doi: 10.1016/j.aml.2016.03.001
[23]	J. Lions, On some equations in boundary value problems of mathematical physics, In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proceedings of the International Symposium, Institute of Mathematics Federal University of Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud., 30 (1978), 284–346.
[24]	J. Liu, A. Hou, J. Liao, Multiplicity of positive solutions for a class of singular elliptic equations with critical Sobolev exponent and Kirchhoff-type nonlocal term, Electron. J. Qual. Theory Differ. Equ., 100 (2018), 1–20.
[25]	X. Liu, Y. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721–733.
[26]	R. Liu, C. Tang, J. Liao, X. Wu, Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four, Commun. Pure Appl. Anal., 15 (2016), 1841–1856. doi: 10.3934/cpaa.2016006
[27]	M. Mu, H. Lu, Existence and multiplicity of positive solutions for Schrödinger-Kirchhoff-Poisson system with singularity, J. Funct. Spaces, 2017 (2017), Article ID 5985962, 12 pages.
[28]	T. Mukherjee, K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differential Equations, 54 (2016), 1–23.
[29]	E. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractioal Sobolev spaces, Bull Sci Math., 136 (2012), 521–573. doi: 10.1016/j.bulsci.2011.12.004
[30]	L. Shao, H. Chen, Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well, C. R. Acad. Sci. Paris, Ser. I, 356 (2018), 489–497. doi: 10.1016/j.crma.2018.03.008
[31]	Y. Sun, Y. Tan, Kirchhoff type equations with strong singularities, Commun. Pure Appl. Anal., 18 (2019), 181–193. doi: 10.3934/cpaa.2019010
[32]	Y. Tang, J. Liao, C. Tang, Two positive solutions for Kirchhoff type problems with Hardy-Sobolev critical exponent and singular nonlinearities, Taiwanese J. Math., 23 (2019), 231–253.
[33]	L. Wang, K. Cheng, B. Zhang, A uniqueness result for strong singular Kirchhoff-type fractional Laplacian problems, Appl. Math. Optim., 2019. Available from: https://doi.org/10.1007/s00245-019-09612-y.
[34]	M. Xiang, B. Zhang, X. Guo, Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem, Nonlinear Anal., 120 (2015), 299–313. doi: 10.1016/j.na.2015.03.015
[35]	S. Yu, J. Chen, Uniqueness and asymptotical behavior of solutions to a Choquard equation with singularity, Appl. Math. Lett., 102 (2020), 106099. doi: 10.1016/j.aml.2019.106099
[36]	S. Yu, J. Chen, Fractional Schrödinger-Poisson system with singularity existence, uniqueness and asymptotic behavior, Glasgow Math. J., 63 (2021), 179–192. doi: 10.1017/S0017089520000099
[37]	S. Yu, J. Chen, Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity, Bound. Value Probl., 30 (2021), 1–18.
[38]	Q. Zhang, Multiple positive solutions for Kirchhoff-Schrödinger-Poisson system with general singularity, Bound. Value Probl., 127 (2017), 1–17.
[39]	Q. Zhang, Existence of positive solution to Kirchhoff-Schrödinger-Poisson system with strong singular term, J. Math. Phys., 60 (2019), 041504. doi: 10.1063/1.5065521

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