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Entire solutions of nonlocal elasticity models for composite materials
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 |
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.
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. |
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H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext. Springer, New York, 2011.
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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. 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.
doi: 10.4171/JEMS/226. |
[7] |
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.
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 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. |
[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,
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[17] |
P. Pucci, X. 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. 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.
doi: 10.4171/JEMS/226. |
[7] |
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.
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 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. |
[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. Pucci, X. 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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