Existence and Multiplicity of Solutions for a Class of Fractional Kirchhoff Type Problems with Variable Exponents
DOI:
https://doi.org/10.15407/mag18.02.253Анотація
У цiй роботi ми розглядаємо певний клас проблем типу Кiрхгофа, якi мiстять дробовий оператор зi змiнними показниками. Використовуючи прямий варiацiйний метод, ми одержуємо результати про iснування розв’язкiв. Крiм того, комбiнуючи теорему про гiрський перевал i варiацiйний принцип Екланда, ми доводимо множиннiсть розв’язкiв. Основнi результати цiєї роботи посилюють i узагальнюють попереднi результати у цiй галузi.
Mathematical Subject Classification 2010: 35J35, 35J60, 46E35
Ключові слова:
дробовий p(x)-лапласiан, варiацiйнi методи, узагальненi простори СоболєваПосилання
E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001), 121-140. https://doi.org/10.1007/s002050100117
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
R. Ayazoglu, Y. Saraç, S. Şule Şener, and G. Alisoy, Existence and multiplicity of solutions for a Schrödinger-Kirchhoff type equation involving the fractional p(•,•)-Laplacian operator in RN, Collect. Math. 72 (2021), 129-156. https://doi.org/10.1007/s13348-020-00283-5
K.B. Ali, M. Hsini, K. Kefi, and N.T. Chung, On a nonlocal fractional p(•,•)-Laplacian with competing nonlinearities, Complex Anal. Oper. Theory 13 (2019),1377-1399. https://doi.org/10.1007/s11785-018-00885-9
K. Ben Ali, M. Bezzarga, A. Ghanmi, and K. Kefi, Existence of positive solution for Kirchhoff problems, Complex Anal. Oper. Theory 13 (2019), 115-126. https://doi.org/10.1007/s11785-017-0709-x
K. Ben Ali, A. Ghanmi, and K. Kefi, Minimax method involving singular p(x)-Kirchhoff equation, J. Math. Phys. 58 (2017), 111505. https://doi.org/10.1063/1.5010798
A. Bahrouni, Comparison and sub-supersolution principles for the fractional p(x)-Laplacian, J. Math. Anal. Appl. 458 (2018), 1363-1372. https://doi.org/10.1016/j.jmaa.2017.10.025
A. Bahrouni and V.D. Rădulescu, On a new fractional Sobolev space and application to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), 379-389. https://doi.org/10.3934/dcdss.2018021
A. Bahrouni, V.D. Rădulescu, and D.D. Repovs, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity 32 (2019), 2481-2495. https://doi.org/10.1088/1361-6544/ab0b03
M. Bouslimi and K. Kefi, Existence of solution for an indefinite weight quasilinear problem with variable exponent, Complex Var. Elliptic Equ. 58 (2013), 1655-1666. https://doi.org/10.1080/17476933.2012.702421
L. Caffarelli, and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260. https://doi.org/10.1080/03605300600987306
R. Chammem, A. Ghanmi, and A. Sahbani, Existence of solution for a singular fractional Laplacian problem with variable exponents and indefinite weights, Complex Var. Elliptic Equ. 66 (2020), 1320-1332. https://doi.org/10.1080/17476933.2020.1756270
M.G. Crandall, P.H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193-222. https://doi.org/10.1080/03605307708820029
L.M. Del Pezzo and J.D. Rossi, Traces for fractional Sobolev spaces with variable exponents, preprint, https://arxiv.org/abs/1704.02599.
D. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), 267-293. https://doi.org/10.4064/sm-143-3-267-293
X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424-446. https://doi.org/10.1006/jmaa.2000.7617
A. Fiscella, A fractional Kirchhoff problem involving a singular term and a critical nonlinearity, Adv. Nonlinear Anal. 8 (2019), 645-660. https://doi.org/10.1515/anona-2017-0075
A. Ghanmi, Nontrivial solutions for Kirchhoff-type problems involving the p(x)-Laplace operator, Rocky Mountain J. Math. 48 (2018), 1145-1158. https://doi.org/10.1216/RMJ-2018-48-4-1145
A. Ghanmi, K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator, Complex Var. Elliptic Equ. 61 (2016), 1199-1216. https://doi.org/10.1080/17476933.2016.1154548
A. Ghanmi and K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fract. Differ. Calc. 6 (2016), 201-217. https://doi.org/10.7153/fdc-06-13
M. Ghergu and V. Radulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, 37, The Clarendon Press, Oxford University Press, Oxford, 2008.
M.K. Hamdani, J. Zuo, N.T. Chung, and D.D. Repovš, Multiplicity of solutions for a class of fractional p(x,•)-Kirchhoff-type problems without the Ambrosetti-Rabinowitz condition, Bound. Value Probl. 2020 (2020), Art. No. 150 https://doi.org/10.1186/s13661-020-01447-9
U. Kaufmann, J.D. Rossi, and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians, preprint, http://mate.dm.uba.ar/~jrossi/krvP.pdf.
K. Kefi, On the existence of solutions of a nonlocal biharmonic problem, Adv. Pureand Appl. Math. 12 (2021), 50-62. https://doi.org/10.21494/ISTE.OP.2020.0581
G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig, 1883.
I.H. Kim, Y.H. Kim, and K. Park, Existence and multiplicity of solutions for Schrödinger-Kirchhoff type problems involving the fractional p(•)-Laplacian in RN, Bound. Value Probl. 2020 (2020), Art. No. 121. https://doi.org/10.4324/9780429340871-6
X. Mingqi, V.D. Radulescu, and B. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Calc. Var. Partial Differential Equations 58 (2019), Art. No. 57. https://doi.org/10.1007/s00526-019-1499-y
N.S. Papageorgiou, V.D. Radulescu, and D.D. Repovš, Positive solutions for nonlinear parametric singular Dirichlet problems, Bull. Math. Sci. 9 (2019), 1950011. https://doi.org/10.1142/S1664360719500115
N.S. Papageorgiou, V.D. Radulescu, D.D. Repovš, Nonlinear nonhomogeneous singular problems, Calc. Var. Partial Differential Equations 59 (2020), Art. No. 9. https://doi.org/10.1007/s00526-019-1667-0
V.D. Radulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. https://doi.org/10.1201/b18601
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, 1748, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000. https://doi.org/10.1007/BFb0104029
R. Servadei and E. Valdinoci, Mountain Pass solutions for nonlocal elliptic operators, J. Math. Anal. Appl. 389 (2012), 887-898. https://doi.org/10.1016/j.jmaa.2011.12.032
R. Servadei and E.Valdinoci, Variational methods for nonlocal operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), 2105-2137. https://doi.org/10.3934/dcds.2013.33.2105
K. Saoudi and A. Ghanmi, A multiplicity results for a singular equation involving the p(x)-Laplace operator, Complex Var. Elliptic Equations 62 (2016), 695-725. https://doi.org/10.1080/17476933.2016.1238466
K. Saoudi, A critical fractional elliptic equation with singular nonlinearities, Fractional Calculus and Applied Analysis 20 (2017), 1-24. https://doi.org/10.1515/fca-2017-0079
W.M. Winslow, Induced fibration of suspensions, J. of Appl. Phys. 20 (1949), 1137-140. https://doi.org/10.1063/1.1698285
M. Xiang, D. Hu, B. Zhang, and Y. Wang, Multiplicity of solutions for variable order fractional Kirchhoff equations with nonstandard growth, J. Math. Anal. Appl. 501 (2020), 124269. https://doi.org/10.1016/j.jmaa.2020.124269
