×

On the orbital stability of fractional Schrödinger equations. (English) Zbl 1282.35319

Summary: 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.

MSC:

35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
47J35 Nonlinear evolution equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. P. Borgna, Existence of ground states for a one dimensional relativistic Schrodinger equations,, J. Math. Phys., 53 (2012) · Zbl 1276.81051 · doi:10.1063/1.4726198
[2] T. Cazenave, <em>Semilinear Schrödinger Equations</em>,, Courant Lecture Notes in Mathematics (2003) · Zbl 1055.35003
[3] Y. Cho, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, Funkcialaj Ekvacioj, 56, 193 (2013) · Zbl 1341.35138
[4] Y. Cho, Global well-posedness of critical nonlinear Schrödinger equations below \(L^2\),, DCDS-A, 33, 1389 (2013) · Zbl 1277.35314 · doi:10.3934/dcds.2013.33.1389
[5] Y. Cho, Strichartz estimates in spherical coordinates,, to appear in Indina Univ. Math. J. · Zbl 1290.42053
[6] Y. Cho, Sobolev inequalities with symmetry,, Contem. Math., 11, 355 (2009) · Zbl 1184.46035 · doi:10.1142/S0219199709003399
[7] Y. Cho, Remarks on some dispersive estimates,, Commun. Pure Appl. Anal., 10, 1121 (2011) · Zbl 1232.35136 · doi:10.3934/cpaa.2011.10.1121
[8] Daoyuan Fang, Weighted strichartz estimates with angular regularity and their applications,, Forum Mathematicum, 23, 181 (2011) · Zbl 1226.35008 · doi:10.1515/FORM.2011.009
[9] P. Felmer, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, P. Roy. Soc. Edinburgh A, 142, 1237 (2012) · Zbl 1290.35308 · doi:10.1017/S0308210511000746
[10] B. Guo, Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations,, J. Math. Phys., 53 (2012) · Zbl 1278.35229 · 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, 17 (2013) · Zbl 1256.49005 · doi:10.1016/j.jmaa.2012.09.023
[12] H. Hajaiej, Necessary and sufficient conditions for the fractional Gargliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, RIMS Kokyuroku Bessatsu, B26, 159 (2011) · Zbl 1270.42026
[13] A. D. Ionescu, Nolinear fractional Schrödinger equations in one dimension,, To appear in J. Funct. Anal. · Zbl 1304.35749 · 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, 109 (1984) · Zbl 0541.49009
[15] T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations,, Cal. Var. PDE., 25, 403 (2006) · Zbl 1089.35071 · doi:10.1007/s00526-005-0349-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.