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An application of Moser's twist theorem to superlinear impulsive differential equations
Normalized solutions of higher-order Schrödinger equations
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
$\mathcal{H}_{0J}u = |u|^{p-2}u+λ u\;\;\;\;{\rm in}\,\,\mathbb{R}^3,\;\;\;\;\;\;\;\;(1)$ |
$\mathcal{H}_{0J}$ |
$J∈\mathbb{N}$ |
$2<p<\frac{4J+6}{3}$ |
$λ∈\mathbb{R}$ |
$E(u)$ |
$E_ρ = \inf\{E(u)|u∈ H^{J}(\mathbb{R}^3):\,\,\|u\|_{L^2(\mathbb{R}^3)} = ρ\}.$ |
$J$ |
$2<p<3$ |
$ρ>0$ |
$2+J<p<\frac{4J+6}{3}$ |
$ρ>0$ |
$J$ |
$3<p<\frac{4J+6}{3}$ |
$ρ>0$ |
References:
[1] |
J. Bellazzini and G. Siciliano,
Stable standing waves for a class of nonlinear SchrödingerPoisson equations, Z. Angew. Math. Phys., 62 (2011), 267-280.
doi: 10.1007/s00033-010-0092-1. |
[2] |
J. Bellazzini and G. Siciliano,
Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.
doi: 10.1016/j.jfa.2011.06.014. |
[3] |
R. Carles, W. Lucha and E. Moulay, Higher-order Schrödinger and Hartree-Fock equations, J. Math. Phys., 56 (2015), 122301, 17 pp.
doi: 10.1063/1.4936646. |
[4] |
R. Carles and E. Moulay, Higher order Schrödinger equations, J. Phys. A, 45 (2012), 395304, 11 pp.
doi: 10.1088/1751-8113/45/39/395304. |
[5] |
X. Chen and J. Yang,
Regularity and symmetry of solutions of an integral equation, Acta Math. Sci., 32 (2012), 1759-1780.
doi: 10.1016/S0252-9602(12)60139-8. |
[6] |
H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Study ed. Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. |
[7] |
P. A. M. Dirac,
The quantum theory of the electron, Proc. R. Soc. A, 117 (1928), 610-624.
|
[8] |
Y. Ebihara and T. Schonbek,
On the (non)compactness of the radial Sobolev spaces, Hiroshima Math. J., 16 (1986), 665-669.
|
[9] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[10] |
A. N. Gorban and I. V. Karlin,
Schrödinger operator in an overfull set, Europhys. Lett., 42 (2007), 113-118.
|
[11] |
R. L. Hall and W. Lucha,
Schrödinger upper bounds to semirelativistic eigenvalues, J. Phys. A, 38 (2005), 7997-8002.
doi: 10.1088/0305-4470/38/37/005. |
[12] |
R. L. Hall and W. Lucha,
Schrödinger secant lower bounds to semirelativistic eigenvalues, Int. J. Mod. Phys. A, 22 (2007), 1899-1904.
doi: 10.1142/S0217751X07036312. |
[13] |
B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics Vol. 1336, Springer-Verlag, Berlin, 1988.
doi: 10.1007/BFb0078115. |
[14] |
Y. Karpeshina and R. Shterenberg, Extended states for polyharmonico perators with quasiperiodic potentials in dimension two, J. Math. Phys., 53 (2012), 103512, 8pp.
doi: 10.1063/1.4754832. |
[15] |
J. M. Kim, A. Arnold and X. Yao,
Global estimates of fundamental solutions for higher-order Schrödinger equations, Monatsh. Math., 168 (2012), 253-266.
doi: 10.1007/s00605-011-0350-0. |
[16] |
E. Lenzmann,
Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.
doi: 10.1007/s11040-007-9020-9. |
[17] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas 14, AMS, 2001.
doi: 10.1090/gsm/014. |
[18] |
P. L. Lions,
The concentration-compactness principle in the Calculus of Variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[19] |
W. Lucha and F. Schöberl, Semirelativistic Bound-State Equations: Trivial Considerations, EPJ Web of Conferences, 80 (2014), 00049. |
[20] |
W. Lucha and F. Schöberl, The spinless relativistic Woods Saxon problem, International Journal of Modern Physics A, 29 (2014), 1450057, 15pp.
doi: 10.1142/S0217751X14500572. |
[21] |
J. Tan, Y. Wang and J. Yang,
Nonlinear Fractional field equations, Nonlinear Anal. TMA, 75 (2012), 2098-2110.
doi: 10.1016/j.na.2011.10.010. |
show all references
References:
[1] |
J. Bellazzini and G. Siciliano,
Stable standing waves for a class of nonlinear SchrödingerPoisson equations, Z. Angew. Math. Phys., 62 (2011), 267-280.
doi: 10.1007/s00033-010-0092-1. |
[2] |
J. Bellazzini and G. Siciliano,
Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.
doi: 10.1016/j.jfa.2011.06.014. |
[3] |
R. Carles, W. Lucha and E. Moulay, Higher-order Schrödinger and Hartree-Fock equations, J. Math. Phys., 56 (2015), 122301, 17 pp.
doi: 10.1063/1.4936646. |
[4] |
R. Carles and E. Moulay, Higher order Schrödinger equations, J. Phys. A, 45 (2012), 395304, 11 pp.
doi: 10.1088/1751-8113/45/39/395304. |
[5] |
X. Chen and J. Yang,
Regularity and symmetry of solutions of an integral equation, Acta Math. Sci., 32 (2012), 1759-1780.
doi: 10.1016/S0252-9602(12)60139-8. |
[6] |
H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Study ed. Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. |
[7] |
P. A. M. Dirac,
The quantum theory of the electron, Proc. R. Soc. A, 117 (1928), 610-624.
|
[8] |
Y. Ebihara and T. Schonbek,
On the (non)compactness of the radial Sobolev spaces, Hiroshima Math. J., 16 (1986), 665-669.
|
[9] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[10] |
A. N. Gorban and I. V. Karlin,
Schrödinger operator in an overfull set, Europhys. Lett., 42 (2007), 113-118.
|
[11] |
R. L. Hall and W. Lucha,
Schrödinger upper bounds to semirelativistic eigenvalues, J. Phys. A, 38 (2005), 7997-8002.
doi: 10.1088/0305-4470/38/37/005. |
[12] |
R. L. Hall and W. Lucha,
Schrödinger secant lower bounds to semirelativistic eigenvalues, Int. J. Mod. Phys. A, 22 (2007), 1899-1904.
doi: 10.1142/S0217751X07036312. |
[13] |
B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics Vol. 1336, Springer-Verlag, Berlin, 1988.
doi: 10.1007/BFb0078115. |
[14] |
Y. Karpeshina and R. Shterenberg, Extended states for polyharmonico perators with quasiperiodic potentials in dimension two, J. Math. Phys., 53 (2012), 103512, 8pp.
doi: 10.1063/1.4754832. |
[15] |
J. M. Kim, A. Arnold and X. Yao,
Global estimates of fundamental solutions for higher-order Schrödinger equations, Monatsh. Math., 168 (2012), 253-266.
doi: 10.1007/s00605-011-0350-0. |
[16] |
E. Lenzmann,
Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.
doi: 10.1007/s11040-007-9020-9. |
[17] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas 14, AMS, 2001.
doi: 10.1090/gsm/014. |
[18] |
P. L. Lions,
The concentration-compactness principle in the Calculus of Variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[19] |
W. Lucha and F. Schöberl, Semirelativistic Bound-State Equations: Trivial Considerations, EPJ Web of Conferences, 80 (2014), 00049. |
[20] |
W. Lucha and F. Schöberl, The spinless relativistic Woods Saxon problem, International Journal of Modern Physics A, 29 (2014), 1450057, 15pp.
doi: 10.1142/S0217751X14500572. |
[21] |
J. Tan, Y. Wang and J. Yang,
Nonlinear Fractional field equations, Nonlinear Anal. TMA, 75 (2012), 2098-2110.
doi: 10.1016/j.na.2011.10.010. |
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