January  2019, 39(1): 447-462. doi: 10.3934/dcds.2019018

Normalized solutions of higher-order Schrödinger equations

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author: Aliang Xia

Received  March 2018 Revised  July 2018 Published  October 2018

Fund Project: The first author is supported by the Foundation of Jiangxi Provincial Education Department, No: GJJ160335, the NNSF of China, No: 11701239 and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. The second author is supported by the NNSF of China, Nos: 11671179 and 11771300

In this paper, we consider the existence of non-trivial solutions for the following equation
$\mathcal{H}_{0J}u = |u|^{p-2}u+λ u\;\;\;\;{\rm in}\,\,\mathbb{R}^3,\;\;\;\;\;\;\;\;(1)$
where
$\mathcal{H}_{0J}$
is the higher-order Schrödinger operator with
$J∈\mathbb{N}$
,
$2<p<\frac{4J+6}{3}$
, and
$λ∈\mathbb{R}$
is a parameter. Let
$E(u)$
be the corresponding variational functional of problem (1). We look for solutions of equation (1) by finding minimizers of the minimization problem
$E_ρ = \inf\{E(u)|u∈ H^{J}(\mathbb{R}^3):\,\,\|u\|_{L^2(\mathbb{R}^3)} = ρ\}.$
We show that problem (1) admits at least a solution provided that in the case
$J$
being odd,
$2<p<3$
and
$ρ>0$
small or
$2+J<p<\frac{4J+6}{3}$
and
$ρ>0$
large; and for the case
$J$
being even,
$3<p<\frac{4J+6}{3}$
and
$ρ>0$
small.
Citation: Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018
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. KimA. 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. TanY. 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. KimA. 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. TanY. 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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