Abstract
This paper presents new multiplicity and existence results for a class of nonlocal fractional problems. Based on critical point theory and the mountain pass theorem, we derive conditions ensuring the multiplicity and existence of weak solution for the fractional problem in a suitable space of functions.
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References
Ali K, Hsini M, Kefi K, Chung N (2019) On a nonlocal fractional \(p (.,.)\)-Laplacian problem with competing nonlinearities. Complex Anal Oper Theory 13(3):1377–1399
Ambrosetti A, Rabinowitz PH (1973) Dual variational methods in critical point theory and applications. J Funct Anal 14(4):349–381
Ayoujil A, El Amrouss A (2009) On the spectrum of a fourth order elliptic equation with variable exponent. Nonlinear Anal Theory Methods Appl 71(10):4916–4926
Azroul E, Boumazourh A (2020) Three solution for a fractional \((p (x,.), q (x,.))\)-Kirchhoff type elliptic system. Nonlinear Funct Anal 2020:Article ID 40
Azroul E, Benkirane A, Shimi M, Srati M (2020) Three solutions for fractional \(p (x,)\)-Laplacian Dirichlet problems with weight. J Nonlinear Funct Anal 2020:1–18
Azroul E, Benkirane A, Boumazourh A, Shimi M (2021) Existence results for fractional p (x,)-Laplacian problem via the Nehari manifold approach. Appl Math Optim 84(2):1527–1547
Bahrouni A, Rădulescu VD (2018) On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin Dyn Syst 11(3):379
Bonanno G (2012) A critical point theorem via the Ekeland variational principle. Nonlinear Anal Theory Methods Appl 75(5):2992–3007
Boureanu MM, Rădulescu V, Repovš D (2016) On a \(p (\cdot )\)-biharmonic problem with no-flux boundary condition. Comput Math Appl 72(9):2505–2515
Caffarelli L (2012) Non-local diffusions, drifts and games. In: Nonlinear partial differential equations, Springer, pp 37–52
Chung NT, Naghizadeh Z (2021) Multiplicity of solutions for a class of fourth-order elliptic equations of \(p(x)\)-Kirchhoff type. Math Slovaca 71(6):1441–1458
Del Pezzo L, Rossi JD (2017) Traces for fractional Sobolev spaces with variable exponents. Adv Oper Theory 2(4):435–446
Di Nezza E, Palatucci G, Valdinoci E (2012) Hitchhiker’s guide to the fractional sobolev spaces. Bull Sci Math 136:521–573
Diening L, Harjulehto P, Hästö P, Ruzicka M (2011) Lebesgue and Sobolev spaces with variable exponents, vol 2017. Springer, New York
Fan X, Zhao D (2001) On the spaces \({L}^{p (x)}({\Omega })\) and \({W}^{m, p (x)}({\Omega })\). J Math Anal Appl 263(2):424–446
Ferrara M, Khademloo S, Heidarkhani S (2014) Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl Math Comput 234:316–325
Hamdani MK, Zuo J, Chung NT (2020) Repovš D (2020) Multiplicity of solutions for a class of fractional \(p (x,) \)-Kirchhoff-type problems without the Ambrosetti-Rabinowitz condition. Bound Value Problem 1:1–16
Jarohs S (2018) Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings. Adv Nonlinear Stud 18(4):691–704
Kaufmann U, Rossi JD, Vidal RE (2017) Fractional Sobolev spaces with variable exponents and fractional \(p (x)\)-Laplacians
Perera K, Squassina M, Yang Y (2016) Bifurcation and multiplicity results for critical fractional p-Laplacian problems. Math Nachr 289(2–3):332–342
Radulescu VD, Repovs DD (2015) Partial differential equations with variable exponents: variational methods and qualitative analysis, vol 9. CRC Press, Boca Raton
Zang A, Fu Y (2008) Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces. Nonlinear Anal Theory Methods Appl 69(10):3629–3636
Zeidler E (2013) Nonlinear functional analysis and its applications: II/B: nonlinear monotone operators. Springer, New York
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The authors would like to thank Associate Editor and three anonymous reviewers for their comments on the manuscript which helped very much in improving and presenting the original version of this paper.
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Communicated by Agnieszka Malinowska.
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Naghizadeh, Z., Nikan, O. & Lopes, A.M. Multiplicity results for a nonlocal fractional problem. Comp. Appl. Math. 41, 239 (2022). https://doi.org/10.1007/s40314-022-01931-1
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DOI: https://doi.org/10.1007/s40314-022-01931-1
Keywords
- Fractional s(x
- .)-Laplacian
- Nonlocal problem
- Fractional Sobolev space with variable exponents
- Critical point theory
- Mountain pass theorem