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Statistical stability for Barge-Martin attractors derived from tent maps
Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems
Mathematics Department, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia |
$ (-\triangle)^{s}_{m}u+V(x)m(u) = f(x,u),\ x\in\mathbb{R}^{d}, $ |
$ 0<s<1 $ |
$ d\geq2 $ |
$ (-\triangle)^{s}_{m} $ |
$ M $ |
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
![]() |
[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. |
[3] |
V. Ambrosio, Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Differential Equations, 2016 (2016), 12 pp. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[25] |
M. A. Krasnosels'kiǐ and J. B. Rutic'kii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd, Groningen, 1961. |
[26] |
A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. |
[27] |
J. Lamperti,
On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459-466.
doi: 10.2140/pjm.1958.8.459. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[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. |
[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. |
[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. |
[43] |
W. M. Zou,
Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.
doi: 10.1007/s002290170032. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
![]() |
[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. |
[3] |
V. Ambrosio, Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Differential Equations, 2016 (2016), 12 pp. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[25] |
M. A. Krasnosels'kiǐ and J. B. Rutic'kii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd, Groningen, 1961. |
[26] |
A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. |
[27] |
J. Lamperti,
On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459-466.
doi: 10.2140/pjm.1958.8.459. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[38] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[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. |
[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. |
[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. |
[43] |
W. M. Zou,
Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.
doi: 10.1007/s002290170032. |
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