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Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential
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Scattering of the focusing energy-critical NLS with inverse square potential in the radial case
Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation
| 1. | Department of Mathematics, ICEx-UFMG, Universidade Federal de Minas Gerais-ICEx, Caixa Postal 702, CEP 30123-970, Belo Horizonte-MG, Brazil |
| 2. | Department of Mathematics, CCN, Universidade Federal do Piauí, Ininga - CEP: 64049-550, Teresina - PI, Brasil |
| $ \begin{equation*} i\partial_{t}u+\Delta u+|x|^{-b}|u|^{p-1}u = 0. \end{equation*} $ |
| $ H^{1}(\mathbb{R^{N}}) $ |
| $ L^{2} $ |
References:
| [1] |
G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2007.
|
| [2] |
A. H. Ardila and V. D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys., 71 (2020), 24pp.
doi: 10.1007/s00033-020-01301-z. |
| [3] |
T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003.
doi: 10.1090/cln/010. |
| [4] |
J. Chen,
On a class of nonlinear inhomogeneous Schrödinger equations, J. Appl. Math. Comput., 32 (2010), 237-253.
doi: 10.1007/s12190-009-0246-5. |
| [5] |
J. Chen and B. Guo,
Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367.
doi: 10.3934/dcdsb.2007.8.357. |
| [6] |
V. Combet and F. Genoud,
Classification of minimal mass blow-up solutions for an ${L}^{2}$ critical inhomogeneous NLS, J. Evol. Equ., 16 (2016), 483-500.
doi: 10.1007/s00028-015-0309-z. |
| [7] |
D. Du, Y. Wu and K. Zhang,
On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin.Dynl. Sys, 36 (2016), 3639-3650.
doi: 10.3934/dcds.2016.36.3639. |
| [8] |
A. de Bouard and R. Fukuizumi,
Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré, 6 (2015), 1157-1177.
doi: 10.1007/s00023-005-0236-6. |
| [9] |
L. G. Farah,
Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.
doi: 10.1007/s00028-015-0298-y. |
| [10] |
G. Fibich and X. P. Wang,
Equations with inhomogeneous nonlinearities, Physica D, 175 (2003), 96-108.
doi: 10.1016/S0167-2789(02)00626-7. |
| [11] |
R. Fukuizumi,
Equations with critical power nonlinearity and potentials., Adv. Differ. Equ., 10 (2005), 259-276.
|
| [12] |
F. Genoud,
Bifurcation and stability of travelling waves in self-focusing planar waveguides, Adv. Nonlinear Stu., 10 (2010), 357-400.
doi: 10.1515/ans-2010-0207. |
| [13] |
F. Genoud,
An inhomogeneous, ${L}^2$-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.
doi: 10.4171/ZAA/1460. |
| [14] |
F. Genoud and C. Stuart,
Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.
doi: 10.3934/dcds.2008.21.137. |
| [15] |
T. Saanouni, Remarks on the inhomogeneous fractional nonlinear Schrödinger equation, J. Math Phys., 57 (2016) 081503.
doi: 10.1063/1.4960045. |
| [16] |
J. Toland,
Uniqueness of positive solutions of some semilinear Sturm-Liouville problemson the half line, Proc. Roy. Soc. Edinburgh Sect. A, 97 (1984), 259-263.
doi: 10.1017/S0308210500032042. |
| [17] |
E. Yanagida,
Uniqueness of positive radial solutions of $\delta u+g(r)u+h(r)u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Rat. Mech. Anal, 115 (1991), 257-274.
doi: 10.1007/BF00380770. |
| [18] |
S. Zhu,
Blow-up solutions for the inhomogeneous Schrödinger equation with ${L}^{2}$ supercritical nonlinearity, J. Math. Anal. Appl., 409 (2014), 760-776.
doi: 10.1016/j.jmaa.2013.07.029. |
show all references
References:
| [1] |
G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2007.
|
| [2] |
A. H. Ardila and V. D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys., 71 (2020), 24pp.
doi: 10.1007/s00033-020-01301-z. |
| [3] |
T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003.
doi: 10.1090/cln/010. |
| [4] |
J. Chen,
On a class of nonlinear inhomogeneous Schrödinger equations, J. Appl. Math. Comput., 32 (2010), 237-253.
doi: 10.1007/s12190-009-0246-5. |
| [5] |
J. Chen and B. Guo,
Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367.
doi: 10.3934/dcdsb.2007.8.357. |
| [6] |
V. Combet and F. Genoud,
Classification of minimal mass blow-up solutions for an ${L}^{2}$ critical inhomogeneous NLS, J. Evol. Equ., 16 (2016), 483-500.
doi: 10.1007/s00028-015-0309-z. |
| [7] |
D. Du, Y. Wu and K. Zhang,
On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin.Dynl. Sys, 36 (2016), 3639-3650.
doi: 10.3934/dcds.2016.36.3639. |
| [8] |
A. de Bouard and R. Fukuizumi,
Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré, 6 (2015), 1157-1177.
doi: 10.1007/s00023-005-0236-6. |
| [9] |
L. G. Farah,
Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.
doi: 10.1007/s00028-015-0298-y. |
| [10] |
G. Fibich and X. P. Wang,
Equations with inhomogeneous nonlinearities, Physica D, 175 (2003), 96-108.
doi: 10.1016/S0167-2789(02)00626-7. |
| [11] |
R. Fukuizumi,
Equations with critical power nonlinearity and potentials., Adv. Differ. Equ., 10 (2005), 259-276.
|
| [12] |
F. Genoud,
Bifurcation and stability of travelling waves in self-focusing planar waveguides, Adv. Nonlinear Stu., 10 (2010), 357-400.
doi: 10.1515/ans-2010-0207. |
| [13] |
F. Genoud,
An inhomogeneous, ${L}^2$-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.
doi: 10.4171/ZAA/1460. |
| [14] |
F. Genoud and C. Stuart,
Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.
doi: 10.3934/dcds.2008.21.137. |
| [15] |
T. Saanouni, Remarks on the inhomogeneous fractional nonlinear Schrödinger equation, J. Math Phys., 57 (2016) 081503.
doi: 10.1063/1.4960045. |
| [16] |
J. Toland,
Uniqueness of positive solutions of some semilinear Sturm-Liouville problemson the half line, Proc. Roy. Soc. Edinburgh Sect. A, 97 (1984), 259-263.
doi: 10.1017/S0308210500032042. |
| [17] |
E. Yanagida,
Uniqueness of positive radial solutions of $\delta u+g(r)u+h(r)u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Rat. Mech. Anal, 115 (1991), 257-274.
doi: 10.1007/BF00380770. |
| [18] |
S. Zhu,
Blow-up solutions for the inhomogeneous Schrödinger equation with ${L}^{2}$ supercritical nonlinearity, J. Math. Anal. Appl., 409 (2014), 760-776.
doi: 10.1016/j.jmaa.2013.07.029. |
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