# American Institute of Mathematical Sciences

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

## On the orbital stability of fractional Schrödinger equations

 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.
Citation: Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [5] Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, to appear in \emph{Indina Univ. Math. J.}, ().   Google Scholar [6] Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Contem. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. D. Ionescu and F. Pusateri, Nolinear fractional Schrödinger equations in one dimension,, \emph{To appear in J. Funct. Anal.}, ().  doi: 10.1016/j.jfa.2013.08.027.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [5] Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, to appear in \emph{Indina Univ. Math. J.}, ().   Google Scholar [6] Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Contem. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. D. Ionescu and F. Pusateri, Nolinear fractional Schrödinger equations in one dimension,, \emph{To appear in J. Funct. Anal.}, ().  doi: 10.1016/j.jfa.2013.08.027.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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