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http://dx.doi.org/10.4134/BKMS.b190029

ON THE ORBITAL STABILITY OF INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATIONS WITH SINGULAR POTENTIAL  

Cho, Yonggeun (Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University)
Lee, Misung (Department of Mathematics Chonbuk National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.6, 2019 , pp. 1601-1615 More about this Journal
Abstract
We show the existence of ground state and orbital stability of standing waves of nonlinear $Schr{\ddot{o}}dinger$ equations with singular linear potential and essentially mass-subcritical power type nonlinearity. For this purpose we establish the existence of ground state in $H^1$. We do not assume symmetry or monotonicity. We also consider local and global well-posedness of Strichartz solutions of energy-subcritical equations. We improve the range of inhomogeneous coefficient in [5, 12] slightly in 3 dimensions.
Keywords
inhomogeneous NLS; singular potential; ground state; orbital stability; well-posedness; Strichartz solution;
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1 A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $R^N$, Rev. Mat. Iberoamericana 6 (1990), no. 1-2, 1-15. https://doi.org/10.4171/RMI/92
2 J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrodinger equations with inhomogeneous nonlinearities, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), no. 2, 221- 233. https://doi.org/10.3934/dcdsb.2008.9.221   DOI
3 T. Cazenave, Semilinear Schrodinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. https://doi.org/10.1090/cln/010
4 Y. Cho, H. Hajaiej, G. Hwang, and T. Ozawa, On the orbital stability of fractional Schrodinger equations, Commun. Pure Appl. Anal. 13 (2014), no. 3, 1267-1282. https: //doi.org/10.3934/cpaa.2014.13.1267   DOI
5 V. D. Dinh, Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrodinger equation, in preprint (arXiv:1710.01392)
6 L. G. Farah and C. M. Guzman, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrodinger equation, J. Differential Equations 262 (2017), no. 8, 4175-4231. https://doi.org/10.1016/j.jde.2017.01.013   DOI
7 R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrodinger equations with inhomogeneous nonlinearities, J. Math. Kyoto Univ. 45 (2005), no. 1, 145-158. https://doi.org/10.1215/kjm/1250282971   DOI
8 F. Genoud, An inhomogeneous, $L^2$-critical, nonlinear Schrodinger equation, Z. Anal. Anwend. 31 (2012), no. 3, 283-290. https://doi.org/10.4171/ZAA/1460   DOI
9 F. Genoud and C. A. Stuart, Schrodinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 137-186. https://doi.org/10.3934/dcds.2008.21.137   DOI
10 T. S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys. 55 (2000), 842-845.
11 C. M. Guzman, On well posedness for the inhomogeneous nonlinear Schrodinger equation, Nonlinear Anal. Real World Appl. 37 (2017), 249-286. https://doi.org/10.1016/j.nonrwa.2017.02.018   DOI
12 M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955-980.   DOI
13 P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), no. 2, 109-145.   DOI
14 C. S. Liu and V. K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasma 1 (9) (1994), 3100-3103.   DOI
15 T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrodinger equations, Calc. Var. Partial Differential Equations 25 (2006), no. 3, 403-408. https://doi.org/10.1007/s00526-005-0349-2   DOI
16 C. Sulem and P.-L. Sulem, The nonlinear Schrodinger Equation, Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999.
17 X.-Y. Tang and P. K. Shukla, Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrodinger equation with an external potential, Physical Review A 76 (2007), 013612-1-10.
18 L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrodinger equation, J. Evol. Equ. 16 (2016), no. 1, 193-208. https://doi.org/10.1007/s00028-015-0298-y   DOI