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

Sabri Bahrouni; Hichem Ounaies

doi:10.3934/dcds.2020155

Previous Article
Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms
DCDS Home
This Issue
Next Article
Statistical stability for Barge-Martin attractors derived from tent maps

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, 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

[1]	Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
[2]	Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236
[3]	Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[4]	Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517
[5]	Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[6]	Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[7]	Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323
[8]	Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[9]	Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential. Electronic Research Archive, , () : -. doi: 10.3934/era.2021047
[10]	Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809
[11]	Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[12]	Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475
[13]	Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[14]	Nguyen Dinh Cong, Doan Thai Son, Stefan Siegmund, Hoang The Tuan. An instability theorem for nonlinear fractional differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3079-3090. doi: 10.3934/dcdsb.2017164
[15]	Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018
[16]	Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402
[17]	Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[18]	Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[19]	Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[20]	Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors