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

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.

Citation: Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
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.

[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.

[3]

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

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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.

[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.

[6]

L. CaffarelliJ.-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.

[7]

L. CaffarelliS. 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.

[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.

[9]

E. Di NezzaG. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

P. PucciX. 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.

[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.

[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.

[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.

[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.

[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.

[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.
[24]

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

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.

[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.

[3]

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

[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.

[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.

[6]

L. CaffarelliJ.-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.

[7]

L. CaffarelliS. 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.

[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.

[9]

E. Di NezzaG. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

P. PucciX. 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.

[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.

[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.

[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.

[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.

[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.

[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.
[24]

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

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