On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)$-Laplacian and without Ambrosetti-Rabinowitz type condition
DOI:
https://doi.org/10.12775/TMNA.2020.072Keywords
Kirchhoff-Choquard equation, Variable order fractional $p(\cdot)$-Laplacian, Fountain theorem, dual fountain theorem, Nehari manifold, Ambrosetti-Rabinowitz type conditionAbstract
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.References
E. Acerbi and G. Mingione, Regularity results for stationary electrorheological fluids, Arch. Rational Mech. Anal. 64 (2002), 213–259.
C.O. Alves, On superlinear p(x)-Laplacian equations in RN , Nonlinear Anal. 73 (2010), 2566–2579.
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.
A. Bahrouni, Comparison and sub-supersolution principles for the fractional p(x)Laplacian, J. Math. Anal. Appl. 458 (2018), no. 2, 1363–1372.
A. Bahrouni and V.D. Rădulescu, 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.
T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), no. 10, 1205–1216.
Z. Binlin, V.D. Rădulescu 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.
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.
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 8, 1245–1260.
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.
F. Dalfovo, S. Giorgini, L. P. Pitaevskiı̆ and S. Stringari, Theory of Bose–Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), no. 3, 463.
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer–Verlag, Heidelberg, 2011.
M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory: the Basis for Linear and Nonlinear Analysis, Springer, New York, 2011.
X. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.
A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170.
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.
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.
K. Ho and Y. H. Kim, A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional p( · )-Laplacian, Nonlinear Anal. 188 (2019), 179–201.
L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on RN , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787–809.
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.
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.
K. Kikuchi and A. Negoro, On Markov processes generated by pseudodifferentail operator of variable order, Osaka J. Math. 34 (1997), 319–335.
H.G. Leopold, Embedding of function spaces of variable order of differentiation, Czechoslovak Math. J. 49 (1999), 633–644.
G. Li, V.D. Rădulescu, D.D. Repovš and Q. Zhang, Nonhomogeneous Dirichlet problems without the Ambrosetti–Rabinowitz condition, Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 55–77.
S. Liang and V.D. Rădulescu, Existence of infinitely many solutions for degenerate Kirchhoff-type Schrodinger–Choquard equations, Electron. J. Differential Equations 2017 (2017), no. 230, 1–17.
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.
P. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), no. 6, 1063–1072.
S. Liu, On ground states of superlinear p-Laplacian equations in RN , J. Math. Anal. Appl. 361 (2010), no. 1, 48–58.
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.
C.F. Lorenzo and T.T. Hartley, Initialized fractional calculus, Int. J. Appl. Math. 3 (2000), 249–265.
C.F. Lorenzo and T.T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam. 29 (2002), 57–98.
D. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal. 99 (2014), 35–48.
G. Molica Bisci, V. D. Rădulescu and R. Servadei, Variational methods for nonlocal fractional problems, vol. 162, Cambridge University Press, 2016.
V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579.
V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005.
T. Mukherjee and K. Sreenadh, Fractional Choquard equation with critical nonlinearities, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 6, 63.
S. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie–Verlag, Berlin, 1954.
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.
P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN , Calc. Var. Part. Differ. Equ. 54 (2015), no. 3, 2785–2806.
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.
P. Pucci, M. Xiang and B. Zhang, Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian, Adv. Calc. Var. 12 (2019), no. 3, 253–275.
V.D. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015), 336–369.
V.D. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, vol. 9, CRC Press, 2015.
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.
S.G. Samko, Convolution and potential type operators in Lp(x) (Rn ), Integral Transforms Spec. Funct. 4 (1998), 261–284.
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.
Z. Tan and F. Fang, On superlinear p(x)-Laplacian problems without Ambrosetti and Rabinowitz condition, Nonlinear Anal. 75 (2012), no. 9, 3902–3915.
M. Willem, Minimax Theorems, vol. 24, Birkhäuser, Boston, 1996.
M. Willem, Functional Analysis: Fundamentals and Applications, vol. 14, Birkhäuser, Basel, 2013.
D. Wu, Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity, J. Math. Anal. Appl. 411 (2014), no. 2, 530–542.
M. Xiang, B. Zhang and V. D. Rădulescu, Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal. 9 (2019), no. 1, 690–709.
M. Xiang, B. Zhang and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal. 178 (2019), 190–204.
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