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May  2014, 13(3): 1267-1282. doi: 10.3934/cpaa.2014.13.1267

On the orbital stability of fractional Schrödinger equations

Yonggeun Cho 1, , Hichem Hajaiej 2, , Gyeongha Hwang 3, and  Tohru Ozawa 4,

1. 

Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2. 

Department of Mathematics, King Saud University, P.O. Box 2455, 11451 Riyadh, Saudi Arabia
3. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea
4. 

Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received  June 2013 Revised  November 2013 Published  December 2013

We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.
Keywords: Fractional Schrödinger equation, finite time blowup., Hartree type nonlinearity, Strichartz estimates.
Mathematics Subject Classification: Primary: 35Q40, 35Q55, 47J3.
Citation: Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267
References:
[1]

J. P. Borgna and D. F. Rial, Existence of ground states for a one dimensional relativistic Schrodinger equations, J. Math. Phys., 53, 062301 (2012). doi: 10.1063/1.4726198.

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.

[3]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcialaj Ekvacioj, 56 (2013), 193-224.

[4]

Y. Cho, G. Hwang and T. Ozawa, Global well-posedness of critical nonlinear Schrödinger equations below $L^2$, DCDS-A, 33 (2013), 1389-1405. doi: 10.3934/dcds.2013.33.1389.

[5]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, to appear in Indina Univ. Math. J., (arXiv:1202.3543v3).

[6]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Contem. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.

[7]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.

[8]

Daoyuan Fang and Chengbo Wang, Weighted strichartz estimates with angular regularity and their applications, Forum Mathematicum, 23 (2011), 181-205. doi: 10.1515/FORM.2011.009.

[9]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, P. Roy. Soc. Edinburgh A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746.

[10]

B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations, J. Math. Phys., 53, 083702 (2012). doi: 10.1063/1.4746806.

[11]

H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity, J. Math. Anal. Appl., 399 (2013), 17-26. doi: 10.1016/j.jmaa.2012.09.023.

[12]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gargliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, RIMS Kokyuroku Bessatsu, B26 (2011), 159-199.

[13]

A. D. Ionescu and F. Pusateri, Nolinear fractional Schrödinger equations in one dimension, To appear in J. Funct. Anal., (arXiv:1209.4943). doi: 10.1016/j.jfa.2013.08.027.

[14]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincare' Anal. Non Line'aire, 1 (1984), 109-145; 223-283.

[15]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Cal. Var. PDE., 25 (2006), 403-408. doi: 10.1007/s00526-005-0349-2.

show all references

References:
[1]

J. P. Borgna and D. F. Rial, Existence of ground states for a one dimensional relativistic Schrodinger equations, J. Math. Phys., 53, 062301 (2012). doi: 10.1063/1.4726198.

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.

[3]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcialaj Ekvacioj, 56 (2013), 193-224.

[4]

Y. Cho, G. Hwang and T. Ozawa, Global well-posedness of critical nonlinear Schrödinger equations below $L^2$, DCDS-A, 33 (2013), 1389-1405. doi: 10.3934/dcds.2013.33.1389.

[5]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, to appear in Indina Univ. Math. J., (arXiv:1202.3543v3).

[6]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Contem. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.

[7]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.

[8]

Daoyuan Fang and Chengbo Wang, Weighted strichartz estimates with angular regularity and their applications, Forum Mathematicum, 23 (2011), 181-205. doi: 10.1515/FORM.2011.009.

[9]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, P. Roy. Soc. Edinburgh A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746.

[10]

B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations, J. Math. Phys., 53, 083702 (2012). doi: 10.1063/1.4746806.

[11]

H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity, J. Math. Anal. Appl., 399 (2013), 17-26. doi: 10.1016/j.jmaa.2012.09.023.

[12]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gargliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, RIMS Kokyuroku Bessatsu, B26 (2011), 159-199.

[13]

A. D. Ionescu and F. Pusateri, Nolinear fractional Schrödinger equations in one dimension, To appear in J. Funct. Anal., (arXiv:1209.4943). doi: 10.1016/j.jfa.2013.08.027.

[14]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincare' Anal. Non Line'aire, 1 (1984), 109-145; 223-283.

[15]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Cal. Var. PDE., 25 (2006), 403-408. doi: 10.1007/s00526-005-0349-2.

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