Multiplicity of solutions to non-local problems of Kirchhoff type involving Hardy potential

Yun-Ho Kim; Hyeon Yeol Na; Yun-Ho Kim; Hyeon Yeol Na

doi:10.3934/math.20231377

AIMS Mathematics

2023, Volume 8, Issue 11: 26896-26921. doi: 10.3934/math.20231377

Previous Article Next Article

Research article Special Issues

Multiplicity of solutions to non-local problems of Kirchhoff type involving Hardy potential

Yun-Ho Kim ^,,
Hyeon Yeol Na

Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea

Received: 28 July 2023 Revised: 11 September 2023 Accepted: 12 September 2023 Published: 21 September 2023
MSC : 35B33, 35D30, 35J20, 35J60, 35J66

The aim of this paper is to establish the existence of a sequence of infinitely many small energy solutions to nonlocal problems of Kirchhoff type involving Hardy potential. To this end, we used the Dual Fountain Theorem as a key tool. In particular, we describe this multiplicity result on a class of the Kirchhoff coefficient and the nonlinear term which differ from previous related works. To the best of our belief, the present paper is the first attempt to obtain the multiplicity result for nonlocal problems of Kirchhoff type involving Hardy potential by utilizing the Dual Fountain Theorem.
- fractional $ p $-Laplacian,
- Kirchhoff function,
- Hardy potential,
- weak solution,
- Dual Fountain Theorem
Citation: Yun-Ho Kim, Hyeon Yeol Na. Multiplicity of solutions to non-local problems of Kirchhoff type involving Hardy potential[J]. AIMS Mathematics, 2023, 8(11): 26896-26921. doi: 10.3934/math.20231377

Related Papers:

Abstract

The aim of this paper is to establish the existence of a sequence of infinitely many small energy solutions to nonlocal problems of Kirchhoff type involving Hardy potential. To this end, we used the Dual Fountain Theorem as a key tool. In particular, we describe this multiplicity result on a class of the Kirchhoff coefficient and the nonlinear term which differ from previous related works. To the best of our belief, the present paper is the first attempt to obtain the multiplicity result for nonlocal problems of Kirchhoff type involving Hardy potential by utilizing the Dual Fountain Theorem.

References

[1]	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. Appl., 41 (2023). https://doi.org/10.1007/s11868-023-00535-5
[2]	R. A. Adams, J. J. F. Fournier, Sobolev spaces, 2 Eds., Academic Press, New York-London, 2003.
[3]	D. Arcoya, J. Carmona, P. J. Martínez-Aparicio, Multiplicity of solutions for an elliptic Kirchhoff equation, Milan J. Math., 90 (2022), 679–689. https://doi.org/10.1007/s00032-022-00365-y doi: 10.1007/s00032-022-00365-y
[4]	G. Autuori, A. Fiscella, P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699–714. https://doi.org/10.1016/j.na.2015.06.014 doi: 10.1016/j.na.2015.06.014
[5]	R. Ayazoglu, S. Akbulut, E. Akkoyunlu, Existence and multiplicity of solutions for $p(.)$-Kirchhoff-type equations, Turkish J. Math., 46 (2022). https://doi.org/10.55730/1300-0098.3164
[6]	B. Barrios, E. Colorado, A. De Pablo, U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differ. Equ., 252 (2012), 6133–6162. https://doi.org/10.1016/j.jde.2012.02.023 doi: 10.1016/j.jde.2012.02.023
[7]	G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal., 54 (2003), 651–665. https://doi.org/10.1016/S0362-546X(03)00092-0 doi: 10.1016/S0362-546X(03)00092-0
[8]	G. Bonanno, S. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1–10. https://doi.org/10.1080/00036810903397438 doi: 10.1080/00036810903397438
[9]	L. Caffarelli, Non-local equations, drifts and games, Nonlinear Partial Differ. Equ. Abel Symp., 7 (2012), 37–52. https://doi.org/10.1007/978-3-642-25361-4
[10]	J. Cen, S. J. Kim, Y. H. Kim, S. Zeng, Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent, Adv. Differential Equ., 28 (2023), 467–504. https://doi.org/10.57262/ade028-0506-467 doi: 10.57262/ade028-0506-467
[11]	G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332–336.
[12]	W. Chen, N. V. Thin, Existence of solutions to Kirchhoff type equations involving the nonlocal $p_1$ & $\cdot\cdot\cdot$ & $p_m$ fractional Laplacian with critical Sobolev-Hardy exponent, Complex Var. Elliptic Equ., 67 (2022), 1931–1975. https://doi.org/10.1080/17476933.2021.1913129 doi: 10.1080/17476933.2021.1913129
[13]	D. Choudhuri, Existence and Hölder regularity of infinitely many solutions to a $p$ Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition, Z. Angew. Math. Phys., 72 (2021). https://doi.org/10.48550/arXiv.2006.00953
[14]	N. T. Chung, H. Q. Toan, On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces, Appl. Math. Comput., 219 (2013), 7820–7829. https://doi.org/10.1016/j.amc.2013.02.011 doi: 10.1016/j.amc.2013.02.011
[15]	G. W. Dai, R. F. Hao, Existence of solutions of a $p(x)$-Kirchhoff-type equation, J. Math. Anal. Appl., 359 (2009), 275–284. https://doi.org/10.1016/j.jmaa.2009.05.031 doi: 10.1016/j.jmaa.2009.05.031
[16]	J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic Equ. Res. Notes Math., 106 (1985).
[17]	J. I. Diaz, J. M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Commun. Part. Diff. Eq., 12 (1987), 1333–1344. https://doi.org/10.1080/03605308708820531 doi: 10.1080/03605308708820531
[18]	M. Fabian, P. Habala, P. Hajék, V. Montesinos, V. Zizler, Banach space theory: The basis for linear and nonlinear analysis, Springer, New York, 2011.
[19]	M. Ferrara, G. M. Bisci, Existence results for elliptic problems with Hardy potential, Bull. Sci. Math., 138 (2014), 846–859. https://doi.org/10.1016/j.bulsci.2014.02.002 doi: 10.1016/j.bulsci.2014.02.002
[20]	A. Fiscella, Schrödinger-Kirchhoff-Hardy $p$-fractional equations without the Ambrosetti-Rabinowitz condition, Discrete Cont. Dyn.-S, 13 (2020), 1993–2007. https://doi.org/10.3934/dcdss.2020154 doi: 10.3934/dcdss.2020154
[21]	A. Fiscella, G. Marino, A. Pinamonti, S. Verzellesi, Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting, Rev. Mat. Complut., 2023, 1–32. https://doi.org/10.1007/s13163-022-00453-y
[22]	A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156–170. https://doi.org/10.1016/j.na.2013.08.011 doi: 10.1016/j.na.2013.08.011
[23]	R. L. 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
[24]	B. Ge, On the superlinear problems involving the $p(x)$-Laplacian and a non-local term without AR-condition, Nonlinear Anal., 102 (2014), 133–143. https://doi.org/10.1016/j.na.2014.02.004 doi: 10.1016/j.na.2014.02.004
[25]	B. Ge, D. J. Lv, J. F. Lu, Multiple solutions for a class of double phase problem without the Ambrosetti-Rabinowitz conditions, Nonlinear Anal., 188 (2019), 294–315. https://doi.org/10.1016/j.na.2019.06.007 doi: 10.1016/j.na.2019.06.007
[26]	G. Gilboa, S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005–1028. https://doi.org/10.1137/070698592 doi: 10.1137/070698592
[27]	S. Gupta, G. Dwivedi, Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces, Math. Method. Appl. Sci., 46 (2023), 8463–8477. https://doi.org/10.1002/mma.8991 doi: 10.1002/mma.8991
[28]	T. Huang, S. Deng, Existence of ground state solutions for Kirchhoff type problem without the Ambrosetti-Rabinowitz condition, Appl. Math. Lett., 113 (2021), 106866. https://doi.org/10.1016/j.aml.2020.106866 doi: 10.1016/j.aml.2020.106866
[29]	E. J. Hurtado, O. H. Miyagaki, R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, J. Dyn. Differ. Equ., 30 (2018), 405–432. https://doi.org/10.1007/s10884-016-9542-6 doi: 10.1007/s10884-016-9542-6
[30]	F. Júlio, S. Corrêa, G. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Aust. Math. Soc., 74 (2006), 263–277. https://doi.org/10.1017/S000497270003570X doi: 10.1017/S000497270003570X
[31]	M. Khodabakhshi, A. M. Aminpour, G. A. Afrouzi, A. Hadjian, Existence of two weak solutions for some singular elliptic problems, RACSAM, 110 (2016), 385–393. https://doi.org/10.1007/s13398-015-0239-1 doi: 10.1007/s13398-015-0239-1
[32]	M. Khodabakhshi, G. A. Afrouzi, A. Hadjian, Existence of infinitely many weak solutions for some singular elliptic problems, Complex Var. Elliptic Equ., 63 (2018), 1570–1580. https://doi.org/10.1080/17476933.2017.1397137 doi: 10.1080/17476933.2017.1397137
[33]	M. Khodabakhshi, A. Hadjian, Existence of three weak solutions for some singular elliptic problems, Complex Var. Elliptic Equ., 63 (2018), 68–75. https://doi.org/10.1080/17476933.2017.1282949 doi: 10.1080/17476933.2017.1282949
[34]	J. M. Kim, Y. H. Kim, Multiple solutions to the double phase problems involving concave-convex nonlinearities, AIMS Math., 8 (2023), 5060–5079. https://doi.org/10.3934/math.2023254 doi: 10.3934/math.2023254
[35]	I. H. Kim, Y. H. Kim, Infinitely many small energy solutions to nonlinear Kirchhoff-Schrödinger equations with the $p$-Laplacian, submitted.
[36]	I. H. Kim, Y. H. Kim, K. Park, Multiple solutions to a non-local problem of Schrödinger-Kirchhoff type in $\Bbb R^{N}$, Fractal Fract., 7 (2023), 627. https://doi.org/10.3390/fractalfract7080627 doi: 10.3390/fractalfract7080627
[37]	G. R. Kirchhoff, Vorlesungen über mathematische physik, mechanik, Teubner, Leipzig, 1876.
[38]	N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000), 298–305. https://doi.org/10.1016/S0375-9601(00)00201-2 doi: 10.1016/S0375-9601(00)00201-2
[39]	J. Lee, J. M. Kim, Y. H. Kim, A. Scapellato, On multiple solutions to a non-local fractional $p(\cdot)$-Laplacian problem with concave-convex nonlinearities, Adv. Cont. Discr. Mod., 2022 (2022), 14. https://doi.org/10.1186/s13662-022-03689-6 doi: 10.1186/s13662-022-03689-6
[40]	G. Li, C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602–4613. https://doi.org/10.1016/j.na.2010.02.037
[41]	L. Li, X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 435 (2016), 955–967. https://doi.org/10.1016/j.jmaa.2015.10.075
[42]	C. B. Lian, B. L. Zhang, B. Ge, Multiple solutions for double phase problems with Hardy type potential, Mathematics, 9 (2021), 376. https://doi.org/10.3390/math9040376 doi: 10.3390/math9040376
[43]	J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284–346. https://doi.org/10.1016/S0304-0208(08)70870-3 doi: 10.1016/S0304-0208(08)70870-3
[44]	D. C. Liu, On a $p$-Kirchhoff-type equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302–308. https://doi.org/10.1016/j.na.2009.06.052 doi: 10.1016/j.na.2009.06.052
[45]	J. Liu, Z. Zhao, Existence of triple solutions for elliptic equations driven by $p$-Laplacian-like operators with Hardy potential under Dirichlet-Neumann boundary conditions, Bound Value Probl., 2023 (2023). https://doi.org/10.1186/s13661-023-01692-8
[46]	S. B. Liu, On superlinear problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788–795. https://doi.org/10.1016/j.na.2010.04.016 doi: 10.1016/j.na.2010.04.016
[47]	S. B. Liu, S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.), 46 (2003), 625–630. https://doi.org/10.12386/A2003sxxb0084 doi: 10.12386/A2003sxxb0084
[48]	D. Lu, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35–48. https://doi.org/10.1016/j.na.2013.12.022 doi: 10.1016/j.na.2013.12.022
[49]	D. Lu, Existence and multiplicity results for perturbed Kirchhoff-type Schrödinger systems in $ {\mathbb R}^3$, Comput. Math. Appl., 68 (2014), 1180–1193. https://doi.org/10.1016/j.camwa.2014.08.020 doi: 10.1016/j.camwa.2014.08.020
[50]	O. H. Miyagaki, M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Diff. Equ., 245 (2008), 3628–3638. https://doi.org/10.1016/j.jde.2008.02.035 doi: 10.1016/j.jde.2008.02.035
[51]	A. Nachman, A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275–281. https://doi.org/10.1137/0138024 doi: 10.1137/0138024
[52]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.48550/arXiv.1104.4345 doi: 10.48550/arXiv.1104.4345
[53]	P. Pucci, S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1–22. https://doi.org/10.4171/RMI/879 doi: 10.4171/RMI/879
[54]	P. Pucci, M. Q. Xiang, B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 54 (2015), 2785–2806. https://doi.org/10.1007/s00526-015-0883-5 doi: 10.1007/s00526-015-0883-5
[55]	P. Pucci, M. Q. Xiang, B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27–55. https://doi.org/10.1515/anona-2015-0102 doi: 10.1515/anona-2015-0102
[56]	B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401–410. https://doi.org/10.1016/S0377-0427(99)00269-1 doi: 10.1016/S0377-0427(99)00269-1
[57]	B. Ricceri, A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151–4157. https://doi.org/10.1016/j.na.2009.02.074 doi: 10.1016/j.na.2009.02.074
[58]	R. Servadei, E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887–898. https://doi.org/10.1016/j.jmaa.2011.12.032 doi: 10.1016/j.jmaa.2011.12.032
[59]	J. Simon, Régularité de la solution d'une équation non linéaire dans $ {\mathbb R}^N$, Journées d'Analyse Non Linéaire, 665 (1978), 205–227. https://doi.org/10.1007/BFb0061807 doi: 10.1007/BFb0061807
[60]	K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\Bbb R^N$, Nonlinear Anal.-Real, 21 (2015), 76–86. https://doi.org/10.1016/j.nonrwa.2014.06.008 doi: 10.1016/j.nonrwa.2014.06.008
[61]	Y. Wei, X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differ. Equ., 52 (2015), 95–124. https://doi.org/10.1007/s00526-013-0706-5 doi: 10.1007/s00526-013-0706-5
[62]	M. Willem, Minimax theorems, Birkhauser, Basel, 1996.
[63]	Q. Wu, X. P. Wu, C. L. Tang, Existence of positive solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, Qual. Theor. Dyn. Syst., 21 (2022), 1–16. https://doi.org/10.1007/s12346-022-00696-6 doi: 10.1007/s12346-022-00696-6
[64]	M. Q. Xiang, B. L. Zhang, X. Y. Guo, Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem, Nonlinear Anal., 120 (2015), 299–313. https://doi.org/10.1016/j.na.2015.03.015 doi: 10.1016/j.na.2015.03.015
[65]	M. Q. Xiang, B. L. Zhang, M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021–1041. https://doi.org/10.1016/j.jmaa.2014.11.055 doi: 10.1016/j.jmaa.2014.11.055
[66]	M. Q. Xiang, B. L. Zhang, M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. R. Soc. A, 471 (2015), 20150034. https://doi.org/10.1098/rspa.2015.0034 doi: 10.1098/rspa.2015.0034
[67]	L. Yang, T. An, Infinitely many solutions for fractional $p$-Kirchhoff equations, Mediterr. J. Math., 15 (2018), 80. https://doi.org/10.1007/s00009-018-1124-x doi: 10.1007/s00009-018-1124-x
[68]	Y. Zhou, J. Wang, L. Zhang, Basic theory of fractional differential equations, 2 Eds., World Scientific Publishing Co. Pte. Ltd., Singapore, 2017.
[69]	J. Zuo, D. Choudhuri, D. D. Repovs, Multiplicity and boundedness of solutions for critical variable-order Kirchhoff type problems involving variable singular exponent, J. Math. Anal. Appl., 514 (2022), 1–18. https://doi.org/10.48550/arXiv.2204.10635 doi: 10.48550/arXiv.2204.10635

Reader Comments

Your name:*

Email:*
© 2023 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)