American Institute of Mathematical Sciences

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June  2018, 11(3): 379-389. doi: 10.3934/dcdss.2018021

On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent

Anouar Bahrouni 1, and  VicenŢiu D. RĂdulescu 2,3,,

1. 

Mathematics Department, University of Monastir, Faculty of Sciences, 5019 Monastir, Tunisia
2. 

Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3. 

Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13,200585 Craiova, Romania

* Corresponding author: Vicent¸iu D. Rădulescu

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: The second author is supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0130.

The content of this paper is at the interplay between function spaces $L^{p(x)}$ and $W^{k, p(x)}$ with variable exponents and fractional Sobolev spaces $W^{s, p}$. We are concerned with some qualitative properties of the fractional Sobolev space $W^{s, q(x), p(x, y)}$, where $q$ and $p$ are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

Keywords: Fractional $p(x)$-Laplace operator, density, integral functional, Gagliardo seminorm, variational method.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35J91, 35S30, 46E35, 58E30.
Citation: Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
References:
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G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^{N}$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar

[2]

A. Bahrouni, Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity, Commun. Pure Appl. Anal., 16 (2017), 243-252.  doi: 10.3934/cpaa.2017011.  Google Scholar

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L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

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L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

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[12]

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. Google Scholar

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P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.  Google Scholar

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G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5.  Google Scholar

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G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397.  Google Scholar

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W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211.  doi: 10.4064/sm-3-1-200-211.  Google Scholar

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P. Pucci, X. Mingqi and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

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V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.  doi: 10.1016/j.na.2014.11.007.  Google Scholar

[19]

V. D. Rădulescu 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. doi: 10.1201/b18601.  Google Scholar

[20]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[21]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[22]

J. Simon, Régularité de la solution d'une équation non linéaire dans ${\mathbb R}^N$, Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), pp. 205-227, Lecture Notes in Math., 665, Springer, Berlin, 1978.  Google Scholar

[23] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.  doi: 10.1007/978-1-4612-0985-0.  Google Scholar
[24]

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.   Google Scholar

show all references

References:
[1]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^{N}$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar

[2]

A. Bahrouni, Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity, Commun. Pure Appl. Anal., 16 (2017), 243-252.  doi: 10.3934/cpaa.2017011.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext. Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[5]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[6]

L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

[7]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[9]

E. Di Nezza, G. Palatucci and 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.  Google Scholar

[10]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ${\mathbb R}^n$ Lecture Notes, Scuola Normale Superiore di Pisa, 15. Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[11]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(Ω)$ and $W^{m, p(x)}(Ω)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[12]

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. Google Scholar

[13]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.  Google Scholar

[14]

G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5.  Google Scholar

[15]

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397.  Google Scholar

[16]

W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200-211.  doi: 10.4064/sm-3-1-200-211.  Google Scholar

[17]

P. Pucci, X. Mingqi and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[18]

V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.  doi: 10.1016/j.na.2014.11.007.  Google Scholar

[19]

V. D. Rădulescu 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. doi: 10.1201/b18601.  Google Scholar

[20]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[21]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[22]

J. Simon, Régularité de la solution d'une équation non linéaire dans ${\mathbb R}^N$, Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), pp. 205-227, Lecture Notes in Math., 665, Springer, Berlin, 1978.  Google Scholar

[23] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.  doi: 10.1007/978-1-4612-0985-0.  Google Scholar
[24]

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.   Google Scholar

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