On the variable exponential fractional Sobolev space Ws(·),p(·)

Haikun Liu; Yongqiang Fu; Haikun Liu; Yongqiang Fu

doi:10.3934/math.2020403

AIMS Mathematics

2020, Volume 5, Issue 6: 6261-6276. doi: 10.3934/math.2020403

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Research article

On the variable exponential fractional Sobolev space W^s(·),p(·)

Haikun Liu ,
Yongqiang Fu ^,

Departnent of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China

Received: 27 May 2020 Accepted: 29 July 2020 Published: 06 August 2020
MSC : 46B20, 46E35, 46B50

In this paper a new kind of variable exponential fractional Sobolev spaces is introduced. For this kind of spaces, some basic properties, such as separability, reflexivity, strict convexity and denseness, are established. At last as an application the existence of solutions for so called s(·)-p(·)- Laplacian equations is discussed.
- variable exponent,
- fractional,
- Sobolev space
Citation: Haikun Liu, Yongqiang Fu. On the variable exponential fractional Sobolev space Ws(·),p(·)[J]. AIMS Mathematics, 2020, 5(6): 6261-6276. doi: 10.3934/math.2020403

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Abstract

In this paper a new kind of variable exponential fractional Sobolev spaces is introduced. For this kind of spaces, some basic properties, such as separability, reflexivity, strict convexity and denseness, are established. At last as an application the existence of solutions for so called s(·)-p(·)- Laplacian equations is discussed.

References

[1]	R. A. Adams, Sobolev Spaces, New York: Academic Press, 1975.
[2]	E. Azroul, A. Benkirane, M. Shimi, Eigenvalue problems involving the fractional p(x)-Laplacian operator, Adv. Oper. Theory, 4 (2019), 539-555. doi: 10.15352/aot.1809-1420
[3]	E. Azroul, A. Benkirane, M. Shimi, et al. On a class of fractional p(x)-Kirchhoff type problems, Appl. Anal., (2019),1-20. Available from: https://www.tandfonline.com/doi/full/10.1080/00036811.2019.1603372
[4]	A. Baalal, M. Berghout, Traces and fractional Sobolev extension domains with variable exponent, Int. J. Math. Anal., 12 (2018), 85-98. doi: 10.12988/ijma.2018.815
[5]	A. Bahrouni, Comparison and sub-supersolution principles for the fractional p(x)-Laplacian, J. Math. Anal. Appl., 458 (2018), 1363-1372. doi: 10.1016/j.jmaa.2017.10.025
[6]	A. Bahrouni, V. Rădulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 379- 389.
[7]	D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces, Foundations and harmonic analysis, Heidelberg: Birkhäuser/Springer, 2013.
[8]	L. M. Del Pezzo, J. D. Rossi, Traces for fractional Sobolev spaces with variable exponents, Adv. Oper. Theory, 2 (2017), 435-446.
[9]	L. Diening, P. Harjulehto, P. Hästö, et al. Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics 2017, Heidelberg: Springer, 2011.
[10]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004
[11]	D. E. Edmunds, J. Lang, A. Nekvinda, On L^p(x) norms, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 219-225. doi: 10.1098/rspa.1999.0309
[12]	D. E. Edmunds, J. Rákosník, Density of smooth functions in W^k,p(x)(Ω), Proc. Roy. Soc. London Ser. A, 437 (1992), 229-236.
[13]	X. L. Fan, D. Zhao, On the spaces spaces L^p(x)(Ω) and W^k,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424-446.
[14]	Y. Q. Fu, X. Zhang, Application of variable exponential function space in partial differential equations (in Chinese), Beijing: Science Press, 2011.
[15]	K. Ho, 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. doi: 10.1016/j.na.2019.06.001
[16]	U. Kaufmann, J. D. Rossi, R. Vidal, Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians, Electron. J. Qual. Theory Differ. Equ., 76 (2017), 1-10.
[17]	O. Kováčik, J. Rákosník, On spaces L^p(x) and W^k,p(x), Czechoslovak Math. J., 41 (1991), 592-618.
[18]	J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Berlin: Springer-Verlag, 1983.
[19]	T. C. Nguyen, Eigenvalue problems for fractional p(x, y)-Laplacian equations with indefinite weight, Taiwanese J. Math., 23 (2019), 1153-1173. doi: 10.11650/tjm/190404
[20]	S. Samko, Denseness of C₀^∞(Rⁿ) in the generalized Sobolev spaces W^m,p(x)(Rⁿ) (Russian), Dokl. Akad. Nauk, 369 (1999), 451-454.
[21]	S. J. Shi, S. T. Chen, Some rotundities musielak-orlicz spaces L^p(x)(Ω) (in Chinese), J. Math., 29 (2009), 211-216.
[22]	I. Singer, On the set of best approximation of an element in a normed linear space, Rev. Math. Pures Appl., 5 (1960), 383-402.
[23]	M. Q. Xiang, B. L. Zhang, D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal., 178 (2019), 190-204. doi: 10.1016/j.na.2018.07.016
[24]	D. J. Guo, Nonlinear functional analysis (in Chinese), Jinan: Shandong Science and Technology Press, 2002.

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