American Institute of Mathematical Sciences

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

Normalized solutions of higher-order Schrödinger equations

Aliang Xia , and  Jianfu Yang

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.
Keywords: Higher-order Schrödinger equations, normalized solutions, existence.
Mathematics Subject Classification: Primary: 35J30, 35J35.
Citation: Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. A, 117 (1928), 610-624.   Google Scholar

[8]

Y. Ebihara and T. Schonbek, On the (non)compactness of the radial Sobolev spaces, Hiroshima Math. J., 16 (1986), 665-669.   Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[10]

A. N. Gorban and I. V. Karlin, Schrödinger operator in an overfull set, Europhys. Lett., 42 (2007), 113-118.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas 14, AMS, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.  Google Scholar

[19]

W. Lucha and F. Schöberl, Semirelativistic Bound-State Equations: Trivial Considerations, EPJ Web of Conferences, 80 (2014), 00049. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. A, 117 (1928), 610-624.   Google Scholar

[8]

Y. Ebihara and T. Schonbek, On the (non)compactness of the radial Sobolev spaces, Hiroshima Math. J., 16 (1986), 665-669.   Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[10]

A. N. Gorban and I. V. Karlin, Schrödinger operator in an overfull set, Europhys. Lett., 42 (2007), 113-118.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas 14, AMS, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.  Google Scholar

[19]

W. Lucha and F. Schöberl, Semirelativistic Bound-State Equations: Trivial Considerations, EPJ Web of Conferences, 80 (2014), 00049. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028
[2]

Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229
[3]

Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part II. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021037
[4]

Ahmet Batal, Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part I. Evolution Equations & Control Theory, 2021, 10 (4) : 861-919. doi: 10.3934/eect.2020095
[5]

Zdeněk Skalák. On the asymptotic decay of higher-order norms of the solutions to the Navier-Stokes equations in R3. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 361-370. doi: 10.3934/dcdss.2010.3.361
[6]

Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051
[7]

Jibin Li, Weigou Rui, Yao Long, Bin He. Travelling wave solutions for higher-order wave equations of KDV type (III). Mathematical Biosciences & Engineering, 2006, 3 (1) : 125-135. doi: 10.3934/mbe.2006.3.125
[8]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[9]

Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3723-3744. doi: 10.3934/cpaa.2021128
[10]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035
[11]

Robert Jankowski, Barbara Łupińska, Magdalena Nockowska-Rosiak, Ewa Schmeidel. Monotonic solutions of a higher-order neutral difference system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 253-261. doi: 10.3934/dcdsb.2018017
[12]

Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789
[13]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[14]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[15]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
[16]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358
[17]

Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771
[18]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911
[19]

Eduard Toon, Pedro Ubilla. Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5831-5843. doi: 10.3934/dcds.2020248
[20]

Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (229)
  • HTML views (150)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy