# American Institute of Mathematical Sciences

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

## Intertwining semiclassical solutions to a Schrödinger-Newton system

 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 $\varepsilon 0.$
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:
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show all references

##### References:
 [1] L. Abatangelo and S. Terracini, Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent,, J. Fixed Point Theory Appl., 10 (2011), 147. doi: 10.1007/s11784-011-0053-0. Google Scholar [2] N. Ackermann, On a periodic Schrödinger equation with nonlocar superlinear part,, Math. Z., 248 (2004), 423. doi: 10.1007/s00209-004-0663-y. Google Scholar [3] S. Cingolani and M. Clapp, Intertwining semiclassical bound states to a nonlinear magnetic equation,, Nonlinearity, 22 (2009), 2309. doi: 10.1088/0951-7715/22/9/013. Google Scholar [4] S. Cingolani and M. Clapp, Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory,, Commun. Pure Appl. Anal., 9 (2010), 1263. doi: 10.3934/cpaa.2010.9.1263. Google Scholar [5] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation,, Z. Angew. Math. Phys. 63 (2012), 63 (2012), 233. doi: 10.1007/s00033-011-0166-8. Google Scholar [6] S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities,, Proc. Roy. Soc. Edinburgh, 140 A (2010), 973. doi: 10.1017/S0308210509000584. Google Scholar [7] M. Clapp and D. Puppe, Critical point theory with symmetries,, J. Reine Angew. Math., 418 (1991), 1. doi: 10.1007/BF02566437. Google Scholar [8] M. Clapp and A. Szulkin, Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential,, Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229. doi: 10.1007/s00030-009-0051-8. Google Scholar [9] T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987). doi: 10.1515/9783110858372.312. Google Scholar [10] J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation,, in, (2004), 2003. Google Scholar [11] J. Fröhlich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation,, Comm. Math. Phys., 225 (2002), 223. doi: 10.1007/s002200100579. Google Scholar [12] R. Harrison, I. Moroz and K. P. Tod, A numerical study of the Schrödinger-Newton equations,, Nonlinearity, 16 (2003), 101. doi: 10.1088/0951-7715/16/1/307. Google Scholar [13] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Stud. Appl. Math., 57 (1977), 93. Google Scholar [14] E. H. Lieb and M. Loss, "Analysis,", Graduate Studies in Math. 14, 14 (1997). Google Scholar [15] P.-L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case,, Ann. Inst. Henry Poincaré, 1 (1984), 109. Google Scholar [16] P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal. T.M.A., 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4. Google Scholar [17] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rational Mech. Anal., 195 (2010), 455. doi: 10.1007/s00205-008-0208-3. Google Scholar [18] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations,, Topology of the Universe Conference (Cleveland, 15 (1998), 2733. doi: 10.1088/0264-9381/15/9/019. Google Scholar [19] I. M. Moroz and P. Tod, An analytical approach to the Schrödinger-Newton equations,, Nonlinearity, 12 (1999), 201. doi: 10.1088/0951-7715/12/2/002. Google Scholar [20] M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential,, Commun. Pure Appl. Anal., 9 (2010), 1411. doi: 10.3934/cpaa.2010.9.1411. Google Scholar [21] R. Palais, The principle of symmetric criticallity,, Comm. Math. Phys., 69 (1979), 19. Google Scholar [22] R. Penrose, On gravity's role in quantum state reduction,, Gen. Rel. Grav., 28 (1996), 581. doi: 10.1007/BF02105068. Google Scholar [23] R. Penrose, Quantum computation, entanglement and state reduction,, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927. doi: 10.1098/rsta.1998.0256. Google Scholar [24] R. Penrose, "The Road to Reality. A Complete Guide to the Laws of the Universe,", Alfred A. Knopf Inc., (2005). Google Scholar [25] S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential,, Nonlinear Analysis, 72 (2010), 3842. doi: 10.1016/j.na.2010.01.021. Google Scholar [26] P. Tod, The ground state energy of the Schrödinger-Newton equation,, Physics Letters A, 280 (2001), 173. doi: 10.1016/S0375-9601(01)00059-7. Google Scholar [27] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3060169. Google Scholar [28] M. Willem, "Minimax Theorems,", PNLDE 24, 24 (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar
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