Basic results of fractional Orlicz-Sobolev space and applications to non-local problems

Sabri Bahrouni, Hichem Ounaies, Leandro S. Tavares


In this paper, we study the interplay between the Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and the fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of a new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is a Young function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of a weak solution for a non-local problem involving the new fractional $M$-Laplacian operator.


Fractional Orlicz-Sobolev space; fractional $M$-Laplacian; non-local problems; existence of solution

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R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in RN, J. Differential Equations 25 (2013), 2340–2362.

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.

A. Bahrouni, Comparison and sub-supersolution principles for the fractional p(x)-Laplacian, J. Math. Anal. Appl. 458 (2018), 1363–1372.

A. Bahrouni and V. Radulescu, On a new fractional Sobolev space and application to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), 379–389.

V. Benci, D. Fortunato and L. Pisani, Solitons like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (1998), 315–344.

J.F. Bonder and A.M. Salort, Fractional order Orlicz–Sobolev spaces, J. Funct. Anal. (2019), DOI: 10.1016/j.jfa.2019.04.003.

H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1992.

Ph. Clément, M. Garcı́a-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. 11 (2000), 33–62.

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016

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.

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.

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of Rn , Lecture Notes, Scuola Normale Superiore di Pisa, vol. 15, Edizioni della Normale, Pisa, 2017.

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer–Verlag, Heidelberg, 2011.

X. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424–446.

N. Fukagai, M. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on RN , Funkc. Ekv. 49 (2006), 235–267.

N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl. 186 (2007), 539–564.

M. Garcı́a-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principal eigenlinear Differential Equations Appl. (NoDEA) 6 (1999), 207–225.

A. Kufner, O. John, S. Fučı́k, Function Spaces, Noordhoff, Leyden, 1977.

U. Kaufmann, J.D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians,∼jrossi/krvP.pdf.

J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459–466.

G. Lieberman, The natural generalizationj of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), 311–361.

M. Mihăilescu and V. Rădulescu, Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces, Anal. Appl. 6 (2008), 1–16.

G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), 2985–3008.

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–212.

M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.

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.

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer–Verlag, Berlin, 2002.

A.M. Salort, A fractional Orlicz–Sobolev eigenvalue problem and related Hardy inequalities, arXiv e-prints (2018), arXiv:1807.03209.

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Sys. 33 (2013), 2105–2137.

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Heidelberg, 1996.

M. Xianga, B. Zhang and D. Yanga, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal. 178 (2019), 190–204.

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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