Existence Results for Kirchhoff Nonlocal Fractional Equations
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Authors: F.-F. LIAO, S. HEIDARKHANI AND A. SALARI
DOI: 10.46793/KgJMat2501.017L
Abstract:
Fractional and nonlocal operators of elliptic type arise in a quite natural way in many different contexts. In this paper, we study the existence of solutions for a class of fractional equations, while the nonlinear part of the problem admits some perturbation property. We obtain some new criteria for existence of two and infinitely many solutions, using critical point theory. Some recent results are extended and improved. Several examples are presented to demonstrate the applications of our main results.
Keywords:
fractional equation, p-Laplacian operator, nonlocal problem, singularity, multiple solutions, critical point theory.
References:
[1] D. Applebaum, Lévy processes-from probalility to finance and quantum groups, Notices Amer. Math. Soc. 51(11) (2004), 1336–1347.
[2] G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ℝN, J. Differential Equations 255(8) (2013), 2340–2362. https://doi.org/10.1016/j.jde.2013.06.016
[3] S. Baraket and G. M. Bisci, Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal. 6(1) (2017), 85–93. https://doi.org/10.1515/anona-2015-0168
[4] B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252(11) (2012), 6133–6162. https://doi.org/10.1016/j.jde.2012.02.023
[5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32(8) (2007), 1245–1260. https://doi.org/10.1080/03605300600987306
[6] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 (2011), 59–88. https://doi.org/10.1007/s00205-010-0336-4
[7] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integrodifferential equations, Comm. Pure Appl. Math. 62(5) (2009), 597–638. https://doi.org/10.1002/cpa.20274
[8] A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Comm. Pure Appl. Anal. 10(6) (2011), 1645–1662. http://dx.doi.org/10.3934/cpaa.2011.10.1645
[9] G. Caristi, S. Heidarkhani, A. Salari and S. A. Tersian, Multiple solutions for degenerate nonlocal problems, Appl. Math. Lett. 84 (2018), 26–33. https://doi.org/10.1016/j.aml.2018.04.007
[10] W. Chen and S. Deng, Existence of solutions for a Kirchhoff type problem involving the fractional p-Laplacian operator, Electron. J. Qual. Theory Differ. Equ. 1(87) (2015), 1–8. http://dx.doi.org/10.14232/ejqtde.2015.1.87
[11] C. Chen and Y. Wei, Existence, nonexistence, and multiple results for the fractional p-Kirchhoff-type equation in , Mediterr. J. Math. 13(6) (2016), 5077–5091. http://dx.doi.org/10.1007/s00009-016-0793-6
[12] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004
[13] S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations 255(1) (2013), 85–119. https://doi.org/10.1016/j.jde.2013.04.001
[14] L. D’Onofrio, V. Ambrosio and G. M. Bisci, Perturbation methods for nonlocal Kirchhoff-type problems, Fract. Calc. Appl. Anal. 20(4) (2017), 829–853. https://doi.org/10.1515/fca-2017-0044
[15] S. Frassu, C. van der Mee and G. Viglialoro, Boundedness in a nonlinear attraction-repulsion Keller-Segel system with production and consumption, J. Math. Anal. Appl. 504(2) (2021), Article ID 125428. https://doi.org/10.1016/j.jmaa.2021.125428
[16] P. Felmer, A. Quaas and J. G. Tan, Positive solutions of nonlinear Schröinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142(6) (2012) 1237-–1262. https://doi.org/10.1017/S0308210511000746
[17] A. Fiscella and 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
[18] G. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, Teubner, Leipzig, 1883.
[19] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268(4–6), 298–305. https://doi.org/10.1016/S0375-9601(00)00201-2
[20] T. Li, N. Pintus and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys. 70(86) (2019), 1–18. https://doi.org/10.1007/s00033-019-1130-2
[21] T. Li and G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differential Integral Equations (2021) (to appear).
[22] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989.
[23] X. Mingqi, G. M. Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity 29 (2016), 357–374. https://doi.org/10.1088/0951-7715
[24] G. M. Bisci, Fractional equations with bounded primitive, Appl. Math. Lett. 27 (2014), 53–58. https://doi.org/10.1016/j.aml.2013.07.011
[25] G. M. Bisci, Sequences of weak solutions for fractional equations, Math. Res. Lett. 21(2) (2014), 241–253. https://dx.doi.org/10.4310/MRL.2014.v21.n2.a3
[26] G. M. Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud. 14(3) (2014), 619–629. https://doi.org/10.1515/ans-2014-0306
[27] G. M. Bisci and D. Repovš, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl. 420(1) (2014), 167–176. https://doi.org/10.1016/j.jmaa.2014.05.073
[28] G. M. Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl. 13(4) (2015), 371–394. https://doi.org/10.1142/S0219530514500067
[29] G. M. Bisci and F. Tulone, An existence result for fractional Kirchhoff-type equations, Z. Anal. Anwend. 35(2) (2016), 181–197. http://dx.doi.org/10.4171/ZAA/1561
[30] Z. Piaoa, C. Zhou and S. Liang, Solutions of stationary Kirchhoff equations involving nonlocal operators with critical nonlinearity in ℝN, Nonlinear Anal. Model. Control 22(5) (2017), 614–635. https://doi.org/10.15388/NA.2017.5.3
[31] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in ℝn involving nonlocal operators, Rev. Mat. Iberoam. 32(1) (2016), 1–22. https://doi.org/10.4171/rmi/879
[32] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, 1986.
[33] B. Ricceri, A multiplicity result for nonlocal problems involving nonlinearities with bounded primitive, Stud. Univ. Babes-Bolyai Math. 4 (2010), 107–114.
[34] R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam. 29(3) (2013), 1091–1126. http://dx.doi.org/10.4171/RMI/750
[35] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389(2) (2012), 887–898. https://doi.org/10.1016/j.jmaa.2011.12.032
[36] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33(5) (2013), 2105–2137. http://dx.doi.org/10.3934/dcds.2013.33.2105
[37] R. Servadei and E. Valdinoci, The Bréezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367(1) (2015), 67–102. https://doi.org/10.1090/S0002-9947-2014-05884-4
[38] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations 42 (2011), 21–41. https://doi.org/10.1007/s00526-010-0378-3
[39] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, In: H. Holden and K. Karlsen (Eds.), Nonlinear Partial Differential Equations, Abel Symposia 7, Springer, Berlin, Heidelberg, 2012, 271–298.
[40] M. Xiang, B. Zhang and V. Rădulescu, Existence of solutions for a bi-nonlocal fractional p-Kirchhoff type problem, Comput. Math. Appl. 71(1) (2016), 255–266. https://doi.org/10.1016/j.camwa.2015.11.017
[41] D. Zhang, Multiple solutions of nonlinear impulsive differential equations with Dirichlet boundary conditions via variational method, Results Math. 63 (2013), 611–628. https://doi.org/10.1007/s00025-011-0221-y
[42] B. Zhang, G. Molica Bisci and M. Xiang, Multiplicity results for nonlocal fractional p-Kirchhoff equations via Morse theory, Topol. Methods Nonlinear Anal. 49(2) (2017), 445–461. https://doi.org/10.12775/TMNA.2016.081