Normalized solutions of higher-order Schrödinger equations

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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 and 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.
[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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