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May  2020, 40(5): 2917-2944. doi: 10.3934/dcds.2020155

Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

Sabri Bahrouni and  Hichem Ounaies

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

Received  August 2019 Published  March 2020

Fund Project: The first and second authors are supported by LR 18 ES 15

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is
$ (-\triangle)^{s}_{m}u+V(x)m(u) = f(x,u),\ x\in\mathbb{R}^{d}, $
where
$ 0<s<1 $
,
$ d\geq2 $
 and
$ (-\triangle)^{s}_{m} $
 is the fractional
$ M $
-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.
Keywords: Fractional Orlicz-Sobolev space, compact embedding theorem, fractional M-Laplacian, fountain Theorem, Schrödinger equation.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35J91, 35S30, 46E35, 58E30.
Citation: Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155
References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.   Google Scholar
[2]

C. O. Alves, G. M. Figueiredo and J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456.  doi: 10.12775/TMNA.2014.055.  Google Scholar

[3]

V. Ambrosio, Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Differential Equations, 2016 (2016), 12 pp.  Google Scholar

[4]

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

[5]

E. Azroul, A. Benkirane and M. Srati, Introduction to fractional Orlicz-Sobolev spaces, arXiv: 1807.11753. Google Scholar

[6]

A. Bahrouni, H. Ounaies and V. D. Rǎdulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.  Google Scholar

[7]

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

[8]

A. Bahrouni, S. Bahrouni and M. Q. Xiang, On a class of nonvariational problems in fractional Orlicz-Sobolev spaces, Nonlinear Analysis, 190 (2020), 111595, 13 pp. doi: 10.1016/j.na.2019.111595.  Google Scholar

[9]

S. Bahrouni, H. Ounaies and L. S. Tavares, Basic results of fractional Orlicz-Sobolev space and applications to non-local problems, Topol. Methods Nonlinear Anal., accepted for publication. Google Scholar

[10]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems in $\mathbb{R}^{d}$, Comm. Partial Differ. Equ., 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[11]

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

[12]

G. Bonanno, G. Molica Bisci and V. Rǎdulescu, Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz-Sobolev spaces, C. R. Math. Acad. Sci. Paris, 349 (2011), 263-268.  doi: 10.1016/j.crma.2011.02.009.  Google Scholar

[13]

J. Fernández Bonder and A. M. Salort, Fractional order Orlicz-Sobolev spaces, Journal of Functional Analysis, 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.  Google Scholar

[14]

J. F. Bonder, M. P. LLanos and A. M. Salort, A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians, arXiv: 1807.01669. Google Scholar

[15]

J. F. Bonder and A. M. Salort, Magnetic Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333–367, arXiv: 1812.05998. doi: 10.1016/j.jfa.2019.04.003.  Google Scholar

[16]

X. J. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Phys., 54 (2013), 061504, 10 pp. doi: 10.1063/1.4809933.  Google Scholar

[17]

Ph. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62.  doi: 10.1007/s005260050002.  Google Scholar

[18]

Ph. Clément, B. de Pagter, G. Sweers and F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math., 1 (2004), 241-267.  doi: 10.1007/s00009-004-0014-6.  Google Scholar

[19]

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

[20]

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

[21]

N. Fukagai, M. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}^{d}$, Funkcialaj Ekvacioj, 49 (2006), 235-267.  doi: 10.1619/fesi.49.235.  Google Scholar

[22]

M. García-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonlinear Differ. Equat. Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.  Google Scholar

[23]

F. Gazzola and V. Rǎdulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbb{R}^{d}$, Differ. Integral Equ., 13 (2000), 47-60.   Google Scholar

[24]

J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Am. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[25]

M. A. Krasnosels'kiǐ and J. B. Rutic'kii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd, Groningen, 1961.  Google Scholar

[26]

A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, Academia, Prague, 1977.  Google Scholar

[27]

J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459-466.  doi: 10.2140/pjm.1958.8.459.  Google Scholar

[28]

M. Mihǎilescu and V. Rǎdulescu, Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces, C. R. Acad. Sci. Paris., 346 (2008), 401-406.  doi: 10.1016/j.crma.2008.02.020.  Google Scholar

[29]

M. Mihǎilescu and V. Rǎdulescu, Existence and multiplicity of solutions for a quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432.  doi: 10.1016/j.jmaa.2006.07.082.  Google Scholar

[30]

M. Mihǎilescu and V. Rǎdulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier, 58 (2008), 2087-2111.  doi: 10.5802/aif.2407.  Google Scholar

[31] G. Molica Bisci, V. D. Rǎdulescu 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
[32]

P. de. Nápoli, J. F. Bonder and A. M. Salort, A Pólya-Szegö principle for general fractional Orlicz-Sobolev spaces, arXiv: 1903.03190. Google Scholar

[33]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[34]

M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.  Google Scholar

[35]

A. M. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, Journal of Differential Equations, (2019). doi: 10.1016/j.jde.2019.11.027.  Google Scholar

[36]

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

[37]

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

[38]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[39]

C. Torres, On superlinear fractional $p$-Laplacian in $\mathbb{R}^{d}$, (2014), arXiv: 1412.3392. Google Scholar

[40]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[41]

Q. Y. Zhang and Q. Wang, Multiple solutions for a class of sublinear Schrödinger equations, J. Math. Anal. Appl., 389 (2012), 511-518.  doi: 10.1016/j.jmaa.2011.12.003.  Google Scholar

[42]

Q. Y. Zhang and B. Xu, Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential, J. Math. Anal. Appl., 377 (2011), 834-840.  doi: 10.1016/j.jmaa.2010.11.059.  Google Scholar

[43]

W. M. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.  doi: 10.1007/s002290170032.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.   Google Scholar
[2]

C. O. Alves, G. M. Figueiredo and J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456.  doi: 10.12775/TMNA.2014.055.  Google Scholar

[3]

V. Ambrosio, Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Differential Equations, 2016 (2016), 12 pp.  Google Scholar

[4]

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

[5]

E. Azroul, A. Benkirane and M. Srati, Introduction to fractional Orlicz-Sobolev spaces, arXiv: 1807.11753. Google Scholar

[6]

A. Bahrouni, H. Ounaies and V. D. Rǎdulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.  Google Scholar

[7]

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

[8]

A. Bahrouni, S. Bahrouni and M. Q. Xiang, On a class of nonvariational problems in fractional Orlicz-Sobolev spaces, Nonlinear Analysis, 190 (2020), 111595, 13 pp. doi: 10.1016/j.na.2019.111595.  Google Scholar

[9]

S. Bahrouni, H. Ounaies and L. S. Tavares, Basic results of fractional Orlicz-Sobolev space and applications to non-local problems, Topol. Methods Nonlinear Anal., accepted for publication. Google Scholar

[10]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems in $\mathbb{R}^{d}$, Comm. Partial Differ. Equ., 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[11]

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

[12]

G. Bonanno, G. Molica Bisci and V. Rǎdulescu, Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz-Sobolev spaces, C. R. Math. Acad. Sci. Paris, 349 (2011), 263-268.  doi: 10.1016/j.crma.2011.02.009.  Google Scholar

[13]

J. Fernández Bonder and A. M. Salort, Fractional order Orlicz-Sobolev spaces, Journal of Functional Analysis, 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.  Google Scholar

[14]

J. F. Bonder, M. P. LLanos and A. M. Salort, A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians, arXiv: 1807.01669. Google Scholar

[15]

J. F. Bonder and A. M. Salort, Magnetic Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333–367, arXiv: 1812.05998. doi: 10.1016/j.jfa.2019.04.003.  Google Scholar

[16]

X. J. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Phys., 54 (2013), 061504, 10 pp. doi: 10.1063/1.4809933.  Google Scholar

[17]

Ph. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33-62.  doi: 10.1007/s005260050002.  Google Scholar

[18]

Ph. Clément, B. de Pagter, G. Sweers and F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math., 1 (2004), 241-267.  doi: 10.1007/s00009-004-0014-6.  Google Scholar

[19]

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

[20]

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

[21]

N. Fukagai, M. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}^{d}$, Funkcialaj Ekvacioj, 49 (2006), 235-267.  doi: 10.1619/fesi.49.235.  Google Scholar

[22]

M. García-Huidobro, V. K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonlinear Differ. Equat. Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.  Google Scholar

[23]

F. Gazzola and V. Rǎdulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in $\mathbb{R}^{d}$, Differ. Integral Equ., 13 (2000), 47-60.   Google Scholar

[24]

J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Am. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.  Google Scholar

[25]

M. A. Krasnosels'kiǐ and J. B. Rutic'kii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd, Groningen, 1961.  Google Scholar

[26]

A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, Academia, Prague, 1977.  Google Scholar

[27]

J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459-466.  doi: 10.2140/pjm.1958.8.459.  Google Scholar

[28]

M. Mihǎilescu and V. Rǎdulescu, Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces, C. R. Acad. Sci. Paris., 346 (2008), 401-406.  doi: 10.1016/j.crma.2008.02.020.  Google Scholar

[29]

M. Mihǎilescu and V. Rǎdulescu, Existence and multiplicity of solutions for a quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432.  doi: 10.1016/j.jmaa.2006.07.082.  Google Scholar

[30]

M. Mihǎilescu and V. Rǎdulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier, 58 (2008), 2087-2111.  doi: 10.5802/aif.2407.  Google Scholar

[31] G. Molica Bisci, V. D. Rǎdulescu 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
[32]

P. de. Nápoli, J. F. Bonder and A. M. Salort, A Pólya-Szegö principle for general fractional Orlicz-Sobolev spaces, arXiv: 1903.03190. Google Scholar

[33]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[34]

M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.  Google Scholar

[35]

A. M. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, Journal of Differential Equations, (2019). doi: 10.1016/j.jde.2019.11.027.  Google Scholar

[36]

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

[37]

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

[38]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[39]

C. Torres, On superlinear fractional $p$-Laplacian in $\mathbb{R}^{d}$, (2014), arXiv: 1412.3392. Google Scholar

[40]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[41]

Q. Y. Zhang and Q. Wang, Multiple solutions for a class of sublinear Schrödinger equations, J. Math. Anal. Appl., 389 (2012), 511-518.  doi: 10.1016/j.jmaa.2011.12.003.  Google Scholar

[42]

Q. Y. Zhang and B. Xu, Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential, J. Math. Anal. Appl., 377 (2011), 834-840.  doi: 10.1016/j.jmaa.2010.11.059.  Google Scholar

[43]

W. M. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.  doi: 10.1007/s002290170032.  Google Scholar

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