Abstract
The main aim of this paper is to investigate the existence of nontrivial solutions for a class of fractional Kirchhoff-type problems with right-hand side nonlinearity which is subcritical or critical exponential growth (subcritical polynomial growth) at infinity. However, it need not satisfy the Ambrosetti–Rabinowitz (AR) condition. Some existence results of nontrivial solutions are established via Mountain Pass Theorem combined with the fractional Moser–Trudinger inequality.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11571176, 11661070, 11764035
Acknowledgements
The author would like to thank the referees for valuable comments and suggestions on improving this paper.
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Author contribution: The author read and approved the final manuscript.
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Research funding: This research is supported by the NSFC (Nos. 11661070, 11764035 and 11571176) and the Science Foundation of Education Department of Gansu (No. 2018B-48).
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Conflict of interest statement: The author declares that he has no competing interest
References
[1] C. O. Alves, F. J. S. A. Corrêa, and T. F. Ma, “Positive solutions for a quasilinear elliptic equation of Kirchhoff type,” Comput. Math. Appl., vol. 49, pp. 85–93, 2005. http://doi.org/10.1016/j.camwa.2005.01.008.10.1016/j.camwa.2005.01.008Search in Google Scholar
[2] G. Autuori, A. Fiscella, and P. Pucci, “Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity,” Nonlinear Anal., vol. 125, pp. 699–714, 2015. https://doi.org/10.1016/j.na.2015.06.014.Search in Google Scholar
[3] A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” J. Funct. Anal., vol. 14, pp. 349–381, 1973. https://doi.org/10.1016/0022-1236(73)90051-7.Search in Google Scholar
[4] V. Ambrosio and T. Isernia, “Concentration phenomena for a fractional Schrödinger–Kirchhoff type equation,” Math. Methods Appl. Sci., vol. 41, pp. 615–645, 2018. https://doi.org/10.1002/mma.4633.Search in Google Scholar
[5] V. Ambrosio and T. Isernia, “A multiplicity result for a fractional Kirchhoff equation in RN${\mathbb{R}}^{N}$ with a general nonlinearity,” Commun. Contemp. Math., vol. 20, p. 1750054, 2018. https://doi.org/10.1142/s0219199717500547.Search in Google Scholar
[6] V. Ambrosio, “On a fractional magnetic Schrödinger equation in R$\mathbb{R}$ with exponential critical growth,” Nonlinear Anal., vol. 183, pp. 117–148, 2019. https://doi.org/10.1016/j.na.2019.01.016.Search in Google Scholar
[7] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana Series, vol. 20, Berlin, Springer-Verlag, 2016.10.1007/978-3-319-28739-3Search in Google Scholar
[8] H. Brezis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,” Comm. Pure Appl. Math., vol. 36, pp. 437–477, 1983. https://doi.org/10.1002/cpa.3160360405.Search in Google Scholar
[9] L. Caffarelli, “Non-local diffusions, drifts and games,” Nonlinear Partial Differential Equations, vol. 7, Abel Symposia, 2012, pp. 37–52.10.1007/978-3-642-25361-4_3Search in Google Scholar
[10] B. Cheng and X. Wu, “Existence results of positive solutions of Kirchhoff-type problems,” Nonlinear Anal., vol. 71, pp. 4883–4892, 2009. https://doi.org/10.1016/j.na.2009.03.065.Search in Google Scholar
[11] B. Cheng, “New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,” J. Math. Anal. Appl., vol. 394, pp. 488–495, 2012. https://doi.org/10.1016/j.jmaa.2012.04.025.Search in Google Scholar
[12] D. G. Costa and O. H. Miyagaki, “Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains,” J. Math. Anal. Appl., vol. 193, pp. 737–755, 1995. https://doi.org/10.1006/jmaa.1995.1264.Search in Google Scholar
[13] E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker’s guide to the fractional Sobolev spaces,” Bull. Sci. Math., vol. 136, pp. 521–573, 2012. https://doi.org/10.1016/j.bulsci.2011.12.004.Search in Google Scholar
[14] D. G. de Figueiredo, J. M. doÓ, and B. Ruf, “Elliptic equations in R2${\mathbb{R}}^{2}$ with nonlinearities in the critical growth range,” Calc. Var. Partial Differ. Equ., vol. 3, pp. 139–153, 1995, http://doi.org/10.1007/bf01205003.10.1007/BF01205003Search in Google Scholar
[15] J. M. doÓ, “Semilinear Dirichlet problems for the N-Laplacian in RN${\mathbb{R}}^{N}$ with nonlinearities in the critical growth range,” Differ. Integr. Equ., vol. 9, pp. 967–979, 1996.10.57262/die/1367871526Search in Google Scholar
[16] J. C. de Albuquerque, Y. L. Araújo, and R. Clemente, “Existence of bound and ground states for a class of Kirchhoff–Schrödinger equations involving critical Trudinger–Moser growth,” Math. Methods Appl. Sci., vol. 42, pp. 806–820, 2019. https://doi.org/10.1002/mma.5382.Search in Google Scholar
[17] G. Devillanova and G. Carlo Marano, “A free fractional viscous oscillator as a forced standard damped vibration,” Frac. Cal. Appl. Anal., vol. 19, pp. 319–356, 2016. https://doi.org/10.1515/fca-2016-0018.Search in Google Scholar
[18] A. Fiscella, G. Molica Bisci, and R. Servadei, “Bifurcation and multiplicity results for critical nonlocal fractional problems,” Bull. Sci. Math., vol. 140, pp. 14–35, 2016. https://doi.org/10.1016/j.bulsci.2015.10.001.Search in Google Scholar
[19] A. Fiscella and E. Valdinoci, “A critical Kirchhoff type problem involving a nonlocal operator,” Nonlinear Anal, vol. 94, pp. 156–170, 2014. https://doi.org/10.1016/j.na.2013.08.011.Search in Google Scholar
[20] X. He and W. Zou, “Infinitely many positive solutions for Kirchhoff-type problems,” Nonlinear Anal., vol. 70, pp. 1407–1414, 2009. https://doi.org/10.1016/j.na.2008.02.011.Search in Google Scholar
[21] A. Iannizzotto and M. Squassina, “1/2-Laplacian problems with exponential nonlinearity,” J. Math. Anal. Appl., vol. 414, pp. 372–385, 2014. https://doi.org/10.1016/j.jmaa.2013.12.059.Search in Google Scholar
[22] T. Isernia, “Sign-changing solutions for a fractional Kirchhoff equation,” Nonlinear Anal., vol. 190, p. 111623, 2020. https://doi.org/10.1016/j.na.2019.111623.Search in Google Scholar
[23] G. Kirchhoff, Mechanik, Leipzig, Teubner, 1883.Search in Google Scholar
[24] N. Lam and G. Z. Lu, “N-Laplacian equations in RN${\mathbb{R}}^{N}$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition,” Adv. Nonlinear Stud., vol. 13, pp. 289–308, 2013. https://doi.org/10.1515/ans-2013-0203.Search in Google Scholar
[25] A. Mao and Z. Zhang, “Sign-changing and multiple solutions of Kirchhoff-type problems without the P. S. condition,” Nonlinear Anal., vol. 70, pp. 1275–1287, 2008. https://doi.org/10.1016/j.na.2008.02.011.Search in Google Scholar
[26] G. M. Bisci, “Sequence of weak solutions for fractional equations,” Math. Res. Lett., vol. 21, pp. 241–253, 2014. https://doi.org/10.4310/mrl.2014.v21.n2.a3.Search in Google Scholar
[27] G. M. Bisci and D. V. Rǎdulescu, “Ground state solutions of scalar field fractional Schrödinger equations,” Calc. Var. Partial Differ. Equ., vol. 54, pp. 2985–3008, 2015. https://doi.org/10.1007/s00526-015-0891-5.Search in Google Scholar
[28] G. M. Bisci, D. V. Rǎdulescu, and R. Servadei, “Variational methods for nonlocal fractional problems,” Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge, Cambridge University Press, 2016.Search in Google Scholar
[29] P. K. Mishra and K. Sreenadh, “Bifurcation and multiplicity of solutions for the fractional Laplacian with critical exponential nonlinearity,” Electron. J. Differ. Equ., vol. 203, pp. 1–9, 2016.Search in Google Scholar
[30] N. Nyamoradi and L. I. Zaidan, “Existence of solutions for degenerate Kirchhoff type problems with fractional p-Laplacian,” Electron. J. Differ. Equ., vol. 115, pp. 1–13, 2017.Search in Google Scholar
[31] N. Nyamoradi and N. T. Chung, “Existence of solutions to nonlocal Kirchhoff equations of elliptic type via genus theory,” Electron. J. Differ. Equ., vol. 86, pp. 1–12, 2014.Search in Google Scholar
[32] N. Nyamoradi and K. M. Teng, “Existence of solutions for a Kirchhoff type nonlocal operators of elliptic type,” Commun. Pure Appl. Anal., vol. 14, pp. 361–371, 2015. https://doi.org/10.3934/cpaa.2015.14.361.Search in Google Scholar
[33] N. Nyamoradi, “Existence of three solutions for Kirchhoff nonlocal operators of elliptic type,” Math. Commun., vol. 18, pp. 489–502, 2013.Search in Google Scholar
[34] E. Parini and B. Ruf, “On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces,” J. Anal. Math., vol. 138, pp. 281–300, 2019. https://doi.org/10.1007/s11854-019-0029-3.Search in Google Scholar
[35] R. Servadei and E. Valdinoci, “Mountain pass solutions for non-local elliptic operators,” J. Math. Anal. Appl., vol. 389, pp. 887–898, 2012. https://doi.org/10.1016/j.jmaa.2011.12.032.Search in Google Scholar
[36] E. Valdinoci, “From the long jump random walk to the fractional Laplacian,” Bol. Soc. Esp. Mat. Apl. Se MA, vol. 49, pp. 33–44, 2009.Search in Google Scholar
[37] M. Q. Xiang, B. L. Zhang, and X. Y. Guo, “Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem,” Nonlinear Anal, vol. 120, pp. 299–313, 2015. https://doi.org/10.1016/j.na.2015.03.015.Search in Google Scholar
[38] M. Q. Xiang, B. L. Zhang, and M. Ferrara, “Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian,” J. Math. Anal. Appl., vol. 424, pp. 1021–1041, 2015. https://doi.org/10.1016/j.jmaa.2014.11.055.Search in Google Scholar
[39] M. Q. Xiang, B. L. Zhang, and M. Ferrara, “Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities,” Proc. Roy. Soc. A, vol. 471, p. 14, 2015. https://doi.org/10.1098/rspa.2015.0034.Search in Google Scholar
[40] M. Q. Xiang, V. D. Rădulescu, and B. L. Zhang, “Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity,” Calc. Var. Partial Differ. Equ., vol. 58, p. 57, 2019. https://doi.org/10.1007/s00526-019-1499-y.Search in Google Scholar
[41] K. Perera and Z. Zhang, “Sign-changing solutions of Kirchhoff-type problems via the Yang index,” J. Differ. Equ., vol. 221, pp. 246–255, 2006. https://doi.org/10.1016/j.jde.2005.03.006.Search in Google Scholar
[42] Z. Zhang and K. Perera, “Sign changing solutions of Kirchhoff-type problems via invariant sets of descent flow,” J. Math. Anal. Appl., vol. 317, pp. 456–463, 2006. https://doi.org/10.1016/j.jmaa.2005.06.102.Search in Google Scholar
[43] Y. M. Zhang and Y. T. Shen, “Existence of solutions for elliptic equations without superquadraticity condition,” Front. Math. China, vol. 7, pp. 587–595, 2012. https://doi.org/10.1007/s11464-012-0211-8.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- Research on bifurcation and chaos characteristics of planet gear transmission system with mixed elastohydrodynamic lubrication (EHL) friction
- Nonlinear modeling and simulation of flywheel energy storage rotor system with looseness and rub-impact coupling hitch
- Application of the piecewise constant method in nonlinear dynamics of drill string
- Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition
- A stabilized fractional-step finite element method for the time-dependent Navier–Stokes equations
- Self-adaptive inertial subgradient extragradient scheme for pseudomonotone variational inequality problem
- An accurate and novel numerical simulation with convergence analysis for nonlinear partial differential equations of Burgers–Fisher type arising in applied sciences
- Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method
- N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions
- Quintic B-spline collocation method for the numerical solution of the Bona–Smith family of Boussinesq equation type
- Existence of positive periodic solutions for a class of second-order neutral functional differential equations