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Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations
On finite Morse index solutions of higher order fractional elliptic equations
1. | Institut Supérieur des Sciences Appliquées et de Technologie de Kairouan, Avenue Beit El Hikma, 3100 Kairouan - Tunisia |
2. | Faculté des Sciences, Département de Mathématiques, B.P 1171 Sfax 3000, Université de Sfax, Tunisia |
$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $ |
$ n\geq 1 $ |
$ \lambda>0 $ |
$ p>1 $ |
$ 1<s\leq2 $ |
$ \lambda u $ |
$ \mathbb R^n $ |
$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}< \frac{\lambda (p+1) }{2} $ |
$ \mathbb R^n $ |
$ p>1 $ |
$ 0<s\leq1 $ |
References:
[1] |
A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992) 1205–1215.
doi: 10.1002/cpa.3160450908. |
[2] |
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. |
[3] |
J. Case and Sun-Yung Alice Chang, On fractional GJMS operators, preprint, arXiv: 1406.1846.
doi: 10.1002/cpa.21564. |
[4] |
W. Chen, X. Cui, R. Zhuo and Z. Yuan, A Liouville theorem for the fractional laplacian, arXiv: 1401.7402. Google Scholar |
[5] |
C. Cowan, P. Esposito and N. Ghoussoub,
Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains, DCDS-A, 28 (2010), 1033-1050.
doi: 10.3934/dcds.2010.28.1033. |
[6] |
C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains, to appear Cal. Var. PDE DOI 10.1007/s00526-012-0582-4.
doi: 10.1007/s00526-012-0582-4. |
[7] |
L. Damascelli and F. Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations, De Gruyter Series in Nonlinear Analysis and Applications, 30, 2019.
doi: 10.1515/9783110538243. |
[8] |
E. N. Dancer,
Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z., 229 (1998), 475-491.
doi: 10.1007/PL00004666. |
[9] |
J. Davila, L. Dupaigne, K. Wang and J. Wei,
A monotonicity formula and a Liouville-type Theorem for a fourth order supercritical problem, Advances in Mathematicas, 258 (2014), 240-285.
doi: 10.1016/j.aim.2014.02.034. |
[10] |
J. Davila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087–6104. Nonlocal time porous medium equation with fractional time derivative. Djida, J.-D., Nieto, J.J., Area, I. Revista Matematica Complutense 32 (2019), 273-304.
doi: 10.1090/tran/6872. |
[11] |
J.D. Djida, J.J. Nieto, I., Area, Nonlocal time porous medium equation with fractional time derivative, Revista Matematica Complutense 32 (2019), 273-304.
doi: 10.1007/s13163-018-0287-0. |
[12] |
X. L. Ding, J.J. Nieto, Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions, Communications in Nonlinear Science and Numerical Simulation 52 (2017), 165-176.
doi: 10.1016/j.cnsns.2017.04.020. |
[13] |
E. B. Fabes, C. E. Kenig, and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116.
doi: 10.1080/03605308208820218. |
[14] |
A. Farina, On the classification of soultions of the Lane-Emden equation on unbounded domains of $ \mathbb R^n$, J. Math. Pures Appl., 87 (9), no. 5,537-561, 2007.
doi: 10.1016/j.matpur.2007.03.001. |
[15] |
M. Fazly, Regularity of extremal solutions of nonlocal elliptic systems, M. Discrete and Continuous Dynamical Systems- Series A 40 (2020), 107-131.
doi: 10.3934/dcds.2020005. |
[16] |
M. Fazly, J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, American Journal of Mathematics, 139, (2), 433-460, 2017.
doi: 10.1353/ajm.2017.0011. |
[17] |
F. Gazzola, H. C. Grunau, Radial entire solutions of supercritical biharmonic equations, Math. Annal., 334, 905-936, 2006.
doi: 10.1007/s00208-005-0748-x. |
[18] |
Z.M. Guo, J. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity, Proc. American Math. Soc. 138, no.11, 3957–3964, 2010.
doi: 10.1090/S0002-9939-10-10374-8. |
[19] |
A.Selmi, A.Harrabi, and C.Zaidi, Nonexistence results on the space or the half space of $-\Delta u+\lambda u = | u|^{p-1} u $ via the Morse index, , Communications on Pure & Applied Analysis 19 (5) (2020): 2839. Google Scholar |
[20] |
A. Harrabi and B. Rahal, On the Sixth-order JosephLundgren Exponent, Journal of Annales Henri Poincaré 18 (3), 1055-1094, 2017.
doi: 10.1007/s00023-016-0522-5. |
[21] |
A. Harrabi and B. Rahal, Liouville type theorems for elliptic equations in half-space with mixed boundary value conditions, J. Advances in Nonlinear Analysis, doi:10.1515/anona-2016-0168 |
[22] |
A. Harrabi and B. Rahal, Liouville results for $m$-Laplace equations in half-space and strips with mixed boundary value conditions and finite Morse index, Journal of Dynamics and Differential Equations, doi: 10.1007/s10884-017-9593-3 |
[23] |
Ira W. Herbst, Spectral theory of the operator $(p^2+m^2)^\frac{1}{2}-Ze^2/r$, , Comm. Math. Phys. 53, no. 3,285-294, 1977. |
[24] |
C. S. Lin, A classification of soluitions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73, 206-231, 1998.
doi: 10.1007/s000140050052. |
[25] |
B. Rahal, C. Zaidi, On the Classification of Stable Solutions of the Fractional Equation, Journal of Potential Anal, (4) 50 (2018): 565-579..
doi: 10.1007/s11118-018-9694-6. |
[26] |
B. Rahal, C. Zaidi, Liouville results for elliptic equations in strips with finite Morse index, Journal of Annali di Matematica https://doi.org/10.1007/s10231-018-0743-y.
doi: 10.1007/s10231-018-0743-y. |
[27] |
X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213, no. 2,587–628, 2014.
doi: 10.1007/s00205-014-0740-2. |
[28] |
J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313, no. 2,207-228, 1999.
doi: 10.1007/s002080050258. |
[29] |
D. Yafaev, Sharp Constants in the Hardy-Rellich Inequalities, Journal of Functional Analysis 168, 121-144, 1999.
doi: 10.1006/jfan.1999.3462. |
[30] |
R. Yang, On higher order extensions for the fractional Laplacian, preprint., arXiv$\sharp$ preprint arXiv: 1302.4413, 2013. Google Scholar |
show all references
References:
[1] |
A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992) 1205–1215.
doi: 10.1002/cpa.3160450908. |
[2] |
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. |
[3] |
J. Case and Sun-Yung Alice Chang, On fractional GJMS operators, preprint, arXiv: 1406.1846.
doi: 10.1002/cpa.21564. |
[4] |
W. Chen, X. Cui, R. Zhuo and Z. Yuan, A Liouville theorem for the fractional laplacian, arXiv: 1401.7402. Google Scholar |
[5] |
C. Cowan, P. Esposito and N. Ghoussoub,
Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains, DCDS-A, 28 (2010), 1033-1050.
doi: 10.3934/dcds.2010.28.1033. |
[6] |
C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains, to appear Cal. Var. PDE DOI 10.1007/s00526-012-0582-4.
doi: 10.1007/s00526-012-0582-4. |
[7] |
L. Damascelli and F. Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations, De Gruyter Series in Nonlinear Analysis and Applications, 30, 2019.
doi: 10.1515/9783110538243. |
[8] |
E. N. Dancer,
Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z., 229 (1998), 475-491.
doi: 10.1007/PL00004666. |
[9] |
J. Davila, L. Dupaigne, K. Wang and J. Wei,
A monotonicity formula and a Liouville-type Theorem for a fourth order supercritical problem, Advances in Mathematicas, 258 (2014), 240-285.
doi: 10.1016/j.aim.2014.02.034. |
[10] |
J. Davila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087–6104. Nonlocal time porous medium equation with fractional time derivative. Djida, J.-D., Nieto, J.J., Area, I. Revista Matematica Complutense 32 (2019), 273-304.
doi: 10.1090/tran/6872. |
[11] |
J.D. Djida, J.J. Nieto, I., Area, Nonlocal time porous medium equation with fractional time derivative, Revista Matematica Complutense 32 (2019), 273-304.
doi: 10.1007/s13163-018-0287-0. |
[12] |
X. L. Ding, J.J. Nieto, Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions, Communications in Nonlinear Science and Numerical Simulation 52 (2017), 165-176.
doi: 10.1016/j.cnsns.2017.04.020. |
[13] |
E. B. Fabes, C. E. Kenig, and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116.
doi: 10.1080/03605308208820218. |
[14] |
A. Farina, On the classification of soultions of the Lane-Emden equation on unbounded domains of $ \mathbb R^n$, J. Math. Pures Appl., 87 (9), no. 5,537-561, 2007.
doi: 10.1016/j.matpur.2007.03.001. |
[15] |
M. Fazly, Regularity of extremal solutions of nonlocal elliptic systems, M. Discrete and Continuous Dynamical Systems- Series A 40 (2020), 107-131.
doi: 10.3934/dcds.2020005. |
[16] |
M. Fazly, J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, American Journal of Mathematics, 139, (2), 433-460, 2017.
doi: 10.1353/ajm.2017.0011. |
[17] |
F. Gazzola, H. C. Grunau, Radial entire solutions of supercritical biharmonic equations, Math. Annal., 334, 905-936, 2006.
doi: 10.1007/s00208-005-0748-x. |
[18] |
Z.M. Guo, J. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity, Proc. American Math. Soc. 138, no.11, 3957–3964, 2010.
doi: 10.1090/S0002-9939-10-10374-8. |
[19] |
A.Selmi, A.Harrabi, and C.Zaidi, Nonexistence results on the space or the half space of $-\Delta u+\lambda u = | u|^{p-1} u $ via the Morse index, , Communications on Pure & Applied Analysis 19 (5) (2020): 2839. Google Scholar |
[20] |
A. Harrabi and B. Rahal, On the Sixth-order JosephLundgren Exponent, Journal of Annales Henri Poincaré 18 (3), 1055-1094, 2017.
doi: 10.1007/s00023-016-0522-5. |
[21] |
A. Harrabi and B. Rahal, Liouville type theorems for elliptic equations in half-space with mixed boundary value conditions, J. Advances in Nonlinear Analysis, doi:10.1515/anona-2016-0168 |
[22] |
A. Harrabi and B. Rahal, Liouville results for $m$-Laplace equations in half-space and strips with mixed boundary value conditions and finite Morse index, Journal of Dynamics and Differential Equations, doi: 10.1007/s10884-017-9593-3 |
[23] |
Ira W. Herbst, Spectral theory of the operator $(p^2+m^2)^\frac{1}{2}-Ze^2/r$, , Comm. Math. Phys. 53, no. 3,285-294, 1977. |
[24] |
C. S. Lin, A classification of soluitions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73, 206-231, 1998.
doi: 10.1007/s000140050052. |
[25] |
B. Rahal, C. Zaidi, On the Classification of Stable Solutions of the Fractional Equation, Journal of Potential Anal, (4) 50 (2018): 565-579..
doi: 10.1007/s11118-018-9694-6. |
[26] |
B. Rahal, C. Zaidi, Liouville results for elliptic equations in strips with finite Morse index, Journal of Annali di Matematica https://doi.org/10.1007/s10231-018-0743-y.
doi: 10.1007/s10231-018-0743-y. |
[27] |
X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213, no. 2,587–628, 2014.
doi: 10.1007/s00205-014-0740-2. |
[28] |
J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313, no. 2,207-228, 1999.
doi: 10.1007/s002080050258. |
[29] |
D. Yafaev, Sharp Constants in the Hardy-Rellich Inequalities, Journal of Functional Analysis 168, 121-144, 1999.
doi: 10.1006/jfan.1999.3462. |
[30] |
R. Yang, On higher order extensions for the fractional Laplacian, preprint., arXiv$\sharp$ preprint arXiv: 1302.4413, 2013. Google Scholar |
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