American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.  Google Scholar

[3]

Nonlinearity, 22 (2009), 2309-2331. doi: 10.1088/0951-7715/22/9/013.  Google Scholar

[4]

Commun. Pure Appl. Anal., 9 (2010), 1263-1281. doi: 10.3934/cpaa.2010.9.1263.  Google Scholar

[5]

Z. Angew. Math. Phys. 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.  Google Scholar

[6]

Proc. Roy. Soc. Edinburgh, 140 A (2010), 973-1009. doi: 10.1017/S0308210509000584.  Google Scholar

[7]

J. Reine Angew. Math., 418 (1991), 1-29. doi: 10.1007/BF02566437.  Google Scholar

[8]

Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229-248. doi: 10.1007/s00030-009-0051-8.  Google Scholar

[9]

Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[10]

in "Séminaire: Équations aux Dérivées Partielles 2003-2004" Exp. No. XIX, 26 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004).  Google Scholar

[11]

Comm. Math. Phys., 225 (2002), 223-274. doi: 10.1007/s002200100579.  Google Scholar

[12]

Nonlinearity, 16 (2003), 101-122. doi: 10.1088/0951-7715/16/1/307.  Google Scholar

[13]

Stud. Appl. Math., 57 (1977), 93-105.  Google Scholar

[14]

Graduate Studies in Math. 14, Amer. Math. Soc. 1997.  Google Scholar

[15]

Ann. Inst. Henry Poincaré, Analyse Non Linéaire, 1 (1984), 109-145 and 223-283. Google Scholar

[16]

Nonlinear Anal. T.M.A., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[17]

Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[18]

Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[19]

Nonlinearity, 12 (1999), 201-216. doi: 10.1088/0951-7715/12/2/002.  Google Scholar

[20]

Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.  Google Scholar

[21]

Comm. Math. Phys., 69 (1979), 19-30.  Google Scholar

[22]

Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.  Google Scholar

[23]

R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.  Google Scholar

[24]

Alfred A. Knopf Inc., New York, 2005.  Google Scholar

[25]

Nonlinear Analysis, 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.  Google Scholar

[26]

Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7.  Google Scholar

[27]

J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.  Google Scholar

[28]

PNLDE 24, Birkhäuser, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

J. Fixed Point Theory Appl., 10 (2011), 147-180. doi: 10.1007/s11784-011-0053-0.  Google Scholar

[2]

Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.  Google Scholar

[3]

Nonlinearity, 22 (2009), 2309-2331. doi: 10.1088/0951-7715/22/9/013.  Google Scholar

[4]

Commun. Pure Appl. Anal., 9 (2010), 1263-1281. doi: 10.3934/cpaa.2010.9.1263.  Google Scholar

[5]

Z. Angew. Math. Phys. 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.  Google Scholar

[6]

Proc. Roy. Soc. Edinburgh, 140 A (2010), 973-1009. doi: 10.1017/S0308210509000584.  Google Scholar

[7]

J. Reine Angew. Math., 418 (1991), 1-29. doi: 10.1007/BF02566437.  Google Scholar

[8]

Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229-248. doi: 10.1007/s00030-009-0051-8.  Google Scholar

[9]

Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[10]

in "Séminaire: Équations aux Dérivées Partielles 2003-2004" Exp. No. XIX, 26 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004).  Google Scholar

[11]

Comm. Math. Phys., 225 (2002), 223-274. doi: 10.1007/s002200100579.  Google Scholar

[12]

Nonlinearity, 16 (2003), 101-122. doi: 10.1088/0951-7715/16/1/307.  Google Scholar

[13]

Stud. Appl. Math., 57 (1977), 93-105.  Google Scholar

[14]

Graduate Studies in Math. 14, Amer. Math. Soc. 1997.  Google Scholar

[15]

Ann. Inst. Henry Poincaré, Analyse Non Linéaire, 1 (1984), 109-145 and 223-283. Google Scholar

[16]

Nonlinear Anal. T.M.A., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[17]

Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[18]

Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[19]

Nonlinearity, 12 (1999), 201-216. doi: 10.1088/0951-7715/12/2/002.  Google Scholar

[20]

Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.  Google Scholar

[21]

Comm. Math. Phys., 69 (1979), 19-30.  Google Scholar

[22]

Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.  Google Scholar

[23]

R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.  Google Scholar

[24]

Alfred A. Knopf Inc., New York, 2005.  Google Scholar

[25]

Nonlinear Analysis, 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.  Google Scholar

[26]

Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7.  Google Scholar

[27]

J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.  Google Scholar

[28]

PNLDE 24, Birkhäuser, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[1]

Kei Nakamura, Tohru Ozawa. Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension. Discrete & 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 & 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 & 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 & Continuous Dynamical Systems - S, 2021, 14 (6) : 1931-1944. 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy