Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

Jie Yang; Lintao Liu; Haibo Chen; Jie Yang; Lintao Liu; Haibo Chen

doi:10.3934/math.2024406

AIMS Mathematics

2024, Volume 9, Issue 4: 8353-8370. doi: 10.3934/math.2024406

Previous Article Next Article

Research article

Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

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

Received: 02 December 2023 Revised: 18 January 2024 Accepted: 29 January 2024 Published: 27 February 2024
MSC : 49J35

This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $

$ \begin{cases} \left(1+\alpha\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s} u+\mu K(x)u = g(x)|u|^{p-2}u, &{\rm in}\ \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \ \end{cases} $

where $ \alpha, \ \mu > 0 $, $ s\in [\frac{3}{4}, 1) $, $ 2 < p < 4 $. By filtration of the Nehari manifold and variational techniques, we obtained the existence of one and two positive solutions under some conditions imposed on $ K $ and $ g $.
- fractional Laplace operator,
- Kirchhoff type problems,
- sign-changing,
- Nehari manifold,
- variational methods
Citation: Jie Yang, Lintao Liu, Haibo Chen. Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions[J]. AIMS Mathematics, 2024, 9(4): 8353-8370. doi: 10.3934/math.2024406

Related Papers:

Abstract

This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $

$ \begin{cases} \left(1+\alpha\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s} u+\mu K(x)u = g(x)|u|^{p-2}u, &{\rm in}\ \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \ \end{cases} $

where $ \alpha, \ \mu > 0 $, $ s\in [\frac{3}{4}, 1) $, $ 2 < p < 4 $. By filtration of the Nehari manifold and variational techniques, we obtained the existence of one and two positive solutions under some conditions imposed on $ K $ and $ g $.

References

[1]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. http://dx.doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
[2]	J. H. He, Q. T. Ain, New promises and future challenges of fractal calculus: From two-scale thermo-dynamics to fractal variational principle, Therm. Sci., 24 (2020), 659–681. http://dx.doi.org/10.2298/TSCI200127065H doi: 10.2298/TSCI200127065H
[3]	T. Bartsch, Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differ. Eq., 20 (1995), 1725–1741.
[4]	C. O. Alves, G. M. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1–26. http://dx.doi.org/10.1515/anona-2015-0101 doi: 10.1515/anona-2015-0101
[5]	T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549–569. http://dx.doi.org/10.1142/S0219199701000494 doi: 10.1142/S0219199701000494
[6]	J. T. Sun, Y. H. Chen, T. F. Wu, Z. S. Feng, Positive solutions of a superlinear Kirchhoff type equation in $\mathbb{R}^{N} (N\geq 4)$, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 141–160. http://dx.doi.org/10.1016/j.cnsns.2018.11.002 doi: 10.1016/j.cnsns.2018.11.002
[7]	J. T. Sun, T. F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differ. Equ., 256 (2014), 1771–1792. http://dx.doi.org/10.1016/j.jde.2013.12.006 doi: 10.1016/j.jde.2013.12.006
[8]	J. T. Sun, T. F. Wu, Steep potential well may help Kirchhoff type equations to generate multiple solutions, Nonlinear Anal., 190 (2020), 111609. http://dx.doi.org/10.1016/j.na.2019.111609 doi: 10.1016/j.na.2019.111609
[9]	L. G. Zhao, H. D. Liu, F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1–23. http://dx.doi.org/10.1016/j.jde.2013.03.005 doi: 10.1016/j.jde.2013.03.005
[10]	F. B. Zhang, M. Du, Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equ., 269 (2020), 10085–10106. http://dx.doi.org/10.1016/j.jde.2020.07.013 doi: 10.1016/j.jde.2020.07.013
[11]	G. F. Che, H. B. Chen, T. F. Wu, Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling, J. Math. Phys., 60 (2019), 081511. http://dx.doi.org/10.1063/1.5087755 doi: 10.1063/1.5087755
[12]	Z. S. Liu, H. J. Luo, Z. T. Zhang, Dancer-Fǔcik spectrum for fractional Schrödinger operators with a steep potential well on $\mathbb{R}^{N}$, Nonlinear Anal., 189 (2019), 111565. http://dx.doi.org/10.1016/j.na.2019.06.024 doi: 10.1016/j.na.2019.06.024
[13]	M. Niu, Z. Tang, Least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian and potential wells, Sci. China Math., 60 (2017), 261–276. https://doi.org/10.1007/s11425-015-0830-3 doi: 10.1007/s11425-015-0830-3
[14]	L. Yang, Z. S. Liu, Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well, Comput. Math. Appl., 72 (2016), 1629–1640. http://dx.doi.org/10.1016/j.camwa.2016.07.033 doi: 10.1016/j.camwa.2016.07.033
[15]	A. Aberqi, A. Ouaziz, Morse's theory and local linking for a fractional $(p_1(x, \cdot), p_2(x, \cdot))$: Laplacian problems on compact manifolds, J. Pseudo-Differ. Oper., 14 (2023). http://dx.doi.org/10.1007/s11868-023-00535-5 doi: 10.1007/s11868-023-00535-5
[16]	Z. Liu, M. Squassina, J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, Nonlinear Differ. Equ. Appl., 24 (2017). http://dx.doi.org/10.1007/s00030-017-0473-7 doi: 10.1007/s00030-017-0473-7
[17]	Z. Liu, V. Rădulescu, Z. Yuan, Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction, Z. Angew. Math. Phys., 73 (2022). http://dx.doi.org/10.1007/s00033-022-01849-y doi: 10.1007/s00033-022-01849-y
[18]	A. Cotsiolis, N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225–236. http://dx.doi.org/10.1016/j.jmaa.2004.03.034 doi: 10.1016/j.jmaa.2004.03.034
[19]	K. J. Brown, Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Eq., 193 (2003), 481–499. http://dx.doi.org/10.1016/S0022-0396(03)00121-9 doi: 10.1016/S0022-0396(03)00121-9
[20]	J. M. Marcos do Ó, X. M. He, P. K. Mishra, Fractional Kirchhoff problem with critical indefinite nonlinearity, Math. Nachr., 292 (2019), 615–632. http://dx.doi.org/10.1002/mana.201800044 doi: 10.1002/mana.201800044
[21]	H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.
[22]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324–353.
[23]	J. Yang, H. B. Chen, Z. S. Feng, Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities, Electron. J. Differ. Eq., 2020 (2020), 1–21.
[24]	B. Abdellaoui, R. Bentifour, Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications, J. Funct. Anal., 272 (2017), 3998–4029. http://dx.doi.org/10.1016/j.jfa.2017.02.007 doi: 10.1016/j.jfa.2017.02.007

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)