Biswas, Reshmi; Tiwari, Sweta On a class of Kirchhoff-Choquard equations involving variable-order fractional \(p(\cdot)\)-Laplacian and without Ambrosetti-Rabinowitz type condition. (English) Zbl 1484.35194 Topol. Methods Nonlinear Anal. 58, No. 2, 403-439 (2021). Summary: In this article, we study the existence of weak solutions and of ground state solutions using the Nehari manifold approach, and existence of infinitely many solutions using the fountain theorem and the dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional \(p(\cdot)\)-Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition. Cited in 5 Documents MSC: 35J60 Nonlinear elliptic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A15 Variational methods applied to PDEs Keywords:Kirchhoff-Choquard equation; existence, variational methods × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] E. Acerbi and G. Mingione, Regularity results for stationary electrorheological fluids, Arch. Rational Mech. Anal. 64 (2002), 213-259. · Zbl 1038.76058 [2] C.O. Alves, On superlinear \(p(x)\)-Laplacian equations in \(\mathbb R^N\), Nonlinear Anal. 73 (2010), 2566-2579. · Zbl 1194.35142 [3] C.O. Alves and L.S. Tavares, A Hardy-Littlewood-Sobolev-type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent, Mediterr. J. Math. 16 (2019), no. 2, 55. · Zbl 1414.35009 [4] A. Bahrouni, Comparison and sub-supersolution principles for the fractional \(p(x)\)-Laplacian, J. Math. Anal. Appl. 458 (2018), no. 2, 1363-1372. · Zbl 1378.35053 [5] A. Bahrouni and V.D. Radulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 379. · Zbl 1374.35158 [6] T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), no. 10, 1205-1216. · Zbl 0799.35071 [7] Z. Binlin, V.D. Radulescu and L. Wang, Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinurgh Sect. A 149 (2019), no. 4, 1061-1081. · Zbl 1442.35501 [8] R. Biswas and S. Tiwari, Variable order nonlocal Choquard problem with variable exponents, Complex Var. Elliptic Equ. (2020), 1-23. DOI: 10.1080/17476933.2020.1751136. · Zbl 1466.35153 [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010. · Zbl 1218.46002 [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 8, 1245-1260. · Zbl 1143.26002 [11] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383-1406. · Zbl 1102.49010 [12] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), no. 3, 463. [13] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. · Zbl 1252.46023 [14] L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Heidelberg, 2011. · Zbl 1222.46002 [15] M. Fabian, P. Habala, P. Hajek, V. Montesinos and V. Zizler, Banach Space Theory: the Basis for Linear and Nonlinear Analysis, Springer, New York, 2011. · Zbl 1229.46001 [16] X. Fan and D. Zhao, On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446. · Zbl 1028.46041 [17] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156-170. · Zbl 1283.35156 [18] Z. Gao, X. Tang and S. Chen, Ground state solutions of fractional Choquard equations with general potentials and nonlinearities, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2037-2057. · Zbl 1418.35361 [19] J. Giacomoni, S. Tiwari and G. Warnault, Quasilinear parabolic problem with \(p(x)\)-Laplacian: existence, uniqueness of weak solutions and stabilization, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 3, 24. · Zbl 1364.35160 [20] K. Ho and Y. H. Kim, A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional \(p( \cdot )\)-Laplacian, Nonlinear Anal. 188 (2019), 179-201. · Zbl 1425.35041 [21] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \(\mathbb R^N\), Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787-809. · Zbl 0935.35044 [22] H. Jin, W. Liu, H. Zhang and J. Zhang, Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth, Commun. Pure Appl. Anal. 19 (2020), no. 1, 123-144. · Zbl 1430.35088 [23] U. Kaufmann, J.D. Rossi and R.E. Vidal, Fractional Sobolev spaces with variable exponents and fractional \(p(x)\)-Laplacians, Electron. J. Qual. Theory Differ. Equ. 76 (2017), 1-10. · Zbl 1413.46035 [24] K. Kikuchi and A. Negoro, On Markov processes generated by pseudodifferentail operator of variable order, Osaka J. Math. 34 (1997), 319-335. · Zbl 0913.60062 [25] H.G. Leopold, Embedding of function spaces of variable order of differentiation, Czechoslovak Math. J. 49 (1999), 633-644. · Zbl 1008.46015 [26] G. Li, V.D. Radulescu, D.D. Repovs and Q. Zhang, Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 55-77. · Zbl 1396.35010 [27] S. Liang and V.D. Radulescu, Existence of infinitely many solutions for degenerate Kirchhoff-type Schrodinger-Choquard equations, Electron. J. Differential Equations 2017 (2017), no. 230, 1-17. · Zbl 1371.35327 [28] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57 (1977), no. 2, 93-105. · Zbl 0369.35022 [29] P. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), no. 6, 1063-1072. · Zbl 0453.47042 [30] S. Liu, On ground states of superlinear \(p\)-Laplacian equations in \(\mathbb R^N\), J. Math. Anal. Appl. 361 (2010), no. 1, 48-58. · Zbl 1178.35174 [31] S. Liu and S.J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (2003), no. 4, 625-630. · Zbl 1081.35043 [32] C.F. Lorenzo and T.T. Hartley, Initialized fractional calculus, Int. J. Appl. Math. 3 (2000), 249-265. · Zbl 1172.26301 [33] C.F. Lorenzo and T.T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam. 29 (2002), 57-98. · Zbl 1018.93007 [34] D. Lu, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal. 99 (2014), 35-48. · Zbl 1286.35108 [35] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational methods for nonlocal fractional problems, vol. 162, Cambridge University Press, 2016. · Zbl 1356.49003 [36] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557-6579. · Zbl 1325.35052 [37] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005. · Zbl 1326.35109 [38] T. Mukherjee and K. Sreenadh, Fractional Choquard equation with critical nonlinearities, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 6, 63. · Zbl 1387.35608 [39] S. Pekar, Untersuchungen uber die Elektronentheorie der Kristalle, Akademie-Verlag, Berlin, 1954. · Zbl 0058.45503 [40] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), no. 1743, 1927-1939. · Zbl 1152.81659 [41] P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrodinger-Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb R^N\), Calc. Var. Part. Differ. Equ. 54 (2015), no. 3, 2785-2806. · Zbl 1329.35338 [42] P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional \(p\)-Kirchhoff equations, Adv. Nonlinear Anal. 5, (2016), no. 1, 27-55. · Zbl 1334.35395 [43] P. Pucci, M. Xiang and B. Zhang, Existence results for Schrodinger-Choquard-Kirchhoff equations involving the fractional \(p\)-Laplacian, Adv. Calc. Var. 12 (2019), no. 3, 253-275. · Zbl 1431.35233 [44] V.D. Radulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015), 336-369. · Zbl 1321.35030 [45] V.D. Radulescu and D. Repovs, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, vol. 9, CRC Press, 2015. · Zbl 1343.35003 [46] M.D. Ruiz-Medina, V.V. Anh and J.M. Angulo, Fractional generalized random fields of variable order, Stoch. Anal. Appl. 22 (2004), 775-799. · Zbl 1069.60040 [47] S.G. Samko, Convolution and potential type operators in \(L^{p(x)}(\mathbb R^n)\), Integral Transforms Spec. Funct. 4 (1998), 261-284. · Zbl 1023.31009 [48] H. Sun, W. Chen, H. Wei and Y.Q. Chen, A comparative study of constantorder and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top. 193 (2011), 185-192. [49] Z. Tan and F. Fang, On superlinear \(p(x)\)-Laplacian problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal. 75 (2012), no. 9, 3902-3915. · Zbl 1241.35047 [50] M. Willem, Minimax Theorems, vol. 24, Birkhauser, Boston, 1996. · Zbl 0856.49001 [51] M. Willem, Functional Analysis: Fundamentals and Applications, vol. 14, Birkhauser, Basel, 2013. · Zbl 1284.46001 [52] D. Wu, Existence and stability of standing waves for nonlinear fractional Schrodinger equations with Hartree type nonlinearity, J. Math. Anal. Appl. 411 (2014), no. 2, 530-542. · Zbl 1332.35343 [53] M. Xiang, B. Zhang and V. D. Radulescu, Superlinear Schrodinger-Kirchhoff type problems involving the fractional \(p\)-Laplacian and critical exponent, Adv. Nonlinear Anal. 9 (2019), no. 1, 690-709. · Zbl 1427.35340 [54] M. Xiang, B. Zhang and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal. 178 (2019), 190-204. · Zbl 1402.35307 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.