Intertwining semiclassical solutions to a Schrödinger-Newton system

Previous Article
Fatigue accumulation in an oscillating plate
DCDS-S Home
This Issue
Next Article
Chaos in forced impact systems

August 2013, 6(4): 891-908. doi: 10.3934/dcdss.2013.6.891

Intertwining semiclassical solutions to a Schrödinger-Newton system

Silvia Cingolani ^1,2, , Mnica Clapp ^3,4, and Simone Secchi ^5,6,7,

1.	Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari
2.	via Orabona 4, 70125 Bari
3.	Instituto de Matemticas, Universidad Nacional Autnoma de Mxico
4.	Circuito Exterior, C.U., 04510 Mxico D.F.
5.	Dipartimento di Matematica ed Applicazioni
6.	Universit di Milano-Bicocca
7.	edificio U5, via R. Cozzi 53, I-20125 Milano

Received October 2011 Revised February 2012 Published December 2012

Abstract
Related Papers

We study the problem\[\begin{cases}\left( -\varepsilon\mathrm{i}\nabla+A(x)\right) ^{2}u+V(x)u=\varepsilon^{-2}\left( \frac{1}{|x|}\ast|u|^{2}\right) u,\\u\in L^{2}(\mathbb{R}^{3},\mathbb{C}), \varepsilon\nablau+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}),\end{cases}\]where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magneticpotential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exteriorelectric potential, and $\varepsilon$ is a small positive number. If $A=0$ and$\varepsilon=\hbar$ is Planck's constant this problem is equivalent to theSchr?dinger-Newton equations proposed by Penrose in [23] todescribe his view that quantum state reduction occurs due to somegravitational effect. We assume that $A$ and $V$ are compatible with theaction of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for anygiven homomorphism $\tau:G\rightarrow\mathbb{S}^{1}$ into the unit complexnumbers, we show that there is a combined effect of the symmetries and thepotential $V$ on the number of semiclassical solutions $u:\mathbb{R}^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=\tau(g)u(x)$ for all $g\in G$,$x\in\mathbb{R}^{3}$. We also study the concentration behavior of thesesolutions as $\varepsilon 0.$

Keywords: Schr?dinger-Newton system, nonlocal nonlinearity, electromagnetic potential, semiclassical solutions, intertwining solutions..

Mathematics Subject Classification: 35Q55, 35Q40, 35J20, 35B0.

Citation: Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891

References:

[1]	J. Fixed Point Theory Appl., 10 (2011), 147-180. doi: 10.1007/s11784-011-0053-0.
[2]	Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.
[3]	Nonlinearity, 22 (2009), 2309-2331. doi: 10.1088/0951-7715/22/9/013.
[4]	Commun. Pure Appl. Anal., 9 (2010), 1263-1281. doi: 10.3934/cpaa.2010.9.1263.
[5]	Z. Angew. Math. Phys. 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.
[6]	Proc. Roy. Soc. Edinburgh, 140 A (2010), 973-1009. doi: 10.1017/S0308210509000584.
[7]	J. Reine Angew. Math., 418 (1991), 1-29. doi: 10.1007/BF02566437.
[8]	Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229-248. doi: 10.1007/s00030-009-0051-8.
[9]	Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.
[10]	in "Séminaire: Équations aux Dérivées Partielles 2003-2004" Exp. No. XIX, 26 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004).
[11]	Comm. Math. Phys., 225 (2002), 223-274. doi: 10.1007/s002200100579.
[12]	Nonlinearity, 16 (2003), 101-122. doi: 10.1088/0951-7715/16/1/307.
[13]	Stud. Appl. Math., 57 (1977), 93-105.
[14]	Graduate Studies in Math. 14, Amer. Math. Soc. 1997.
[15]	Ann. Inst. Henry Poincaré, Analyse Non Linéaire, 1 (1984), 109-145 and 223-283.
[16]	Nonlinear Anal. T.M.A., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.
[17]	Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.
[18]	Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.
[19]	Nonlinearity, 12 (1999), 201-216. doi: 10.1088/0951-7715/12/2/002.
[20]	Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.
[21]	Comm. Math. Phys., 69 (1979), 19-30.
[22]	Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.
[23]	R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.
[24]	Alfred A. Knopf Inc., New York, 2005.
[25]	Nonlinear Analysis, 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.
[26]	Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7.
[27]	J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.
[28]	PNLDE 24, Birkhäuser, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:

[1]	J. Fixed Point Theory Appl., 10 (2011), 147-180. doi: 10.1007/s11784-011-0053-0.
[2]	Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.
[3]	Nonlinearity, 22 (2009), 2309-2331. doi: 10.1088/0951-7715/22/9/013.
[4]	Commun. Pure Appl. Anal., 9 (2010), 1263-1281. doi: 10.3934/cpaa.2010.9.1263.
[5]	Z. Angew. Math. Phys. 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.
[6]	Proc. Roy. Soc. Edinburgh, 140 A (2010), 973-1009. doi: 10.1017/S0308210509000584.
[7]	J. Reine Angew. Math., 418 (1991), 1-29. doi: 10.1007/BF02566437.
[8]	Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229-248. doi: 10.1007/s00030-009-0051-8.
[9]	Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.
[10]	in "Séminaire: Équations aux Dérivées Partielles 2003-2004" Exp. No. XIX, 26 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004).
[11]	Comm. Math. Phys., 225 (2002), 223-274. doi: 10.1007/s002200100579.
[12]	Nonlinearity, 16 (2003), 101-122. doi: 10.1088/0951-7715/16/1/307.
[13]	Stud. Appl. Math., 57 (1977), 93-105.
[14]	Graduate Studies in Math. 14, Amer. Math. Soc. 1997.
[15]	Ann. Inst. Henry Poincaré, Analyse Non Linéaire, 1 (1984), 109-145 and 223-283.
[16]	Nonlinear Anal. T.M.A., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.
[17]	Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.
[18]	Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.
[19]	Nonlinearity, 12 (1999), 201-216. doi: 10.1088/0951-7715/12/2/002.
[20]	Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.
[21]	Comm. Math. Phys., 69 (1979), 19-30.
[22]	Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.
[23]	R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.
[24]	Alfred A. Knopf Inc., New York, 2005.
[25]	Nonlinear Analysis, 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.
[26]	Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7.
[27]	J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.
[28]	PNLDE 24, Birkhäuser, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[1]	Kei Nakamura, Tohru Ozawa. Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 789-801. doi: 10.3934/dcds.2013.33.789
[2]	Jing Yang. Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1785-1805. doi: 10.3934/cpaa.2017087
[3]	Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[4]	Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021016
[5]	Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
[6]	Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[7]	Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[8]	Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125
[9]	Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1967-1981. doi: 10.3934/dcdss.2021008
[10]	Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3723-3744. doi: 10.3934/cpaa.2021128
[11]	Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109
[12]	Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure and Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771
[13]	Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831
[14]	Thomas Bartsch, Zhongwei Tang. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 7-26. doi: 10.3934/dcds.2013.33.7
[15]	Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[16]	Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025
[17]	Dengfeng Lü. Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity. Communications on Pure and Applied Analysis, 2018, 17 (2) : 605-626. doi: 10.3934/cpaa.2018033
[18]	J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445
[19]	Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292
[20]	Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Silvia Cingolani Mnica Clapp Simone Secchi
on Google Scholar
Silvia Cingolani Mnica Clapp Simone Secchi

[Back to Top]