Bonheure, Denis; Cingolani, Silvia; Secchi, Simone Concentration phenomena for the Schrödinger-Poisson system in \(\mathbb{R}^2\). (English) Zbl 1480.35153 Discrete Contin. Dyn. Syst., Ser. S 14, No. 5, 1631-1648 (2021). Summary: We perform a semiclassical analysis for the planar Schrödinger-Poisson system \[ \begin{cases} -\varepsilon^2 \Delta\psi+V(x)\psi = E(x) \psi \quad \text{in }\mathbb{R}^2, \\ -\Delta E = |\psi|^2 \quad \text{in }\mathbb{R}^2, \end{cases} \eqno{(SP_\varepsilon)} \] where \(\varepsilon\) is a positive parameter corresponding to the Planck constant and \(V\) is a bounded external potential. We detect solution pairs \((u_\varepsilon, E_\varepsilon)\) of the system \((SP_\varepsilon)\) as \(\varepsilon \rightarrow 0 \), leaning on a nongeneracy result in [the first author et al., J. Funct. Anal. 272, No. 12, 5255–5281 (2017; Zbl 1386.35052)]. Cited in 7 Documents MSC: 35J47 Second-order elliptic systems 35J61 Semilinear elliptic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 35A01 Existence problems for PDEs: global existence, local existence, non-existence Keywords:Schrödinger-Poisson system; nonlocal nonlinearity; logarithmic potential; existence Citations:Zbl 1386.35052 PDFBibTeX XMLCite \textit{D. Bonheure} et al., Discrete Contin. Dyn. Syst., Ser. S 14, No. 5, 1631--1648 (2021; Zbl 1480.35153) Full Text: DOI arXiv References: [1] A. Ambrosetti; M. Badiale; S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140, 285-300 (1997) · Zbl 0896.35042 [2] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on \(\begin{document} \mathbb{R}^n\end{document}} \), Birkhäuser Verlag, 2006. · Zbl 1115.35004 [3] D. Bonheure; S. Cingolani; J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272, 5255-5281 (2017) · Zbl 1386.35052 [4] P. Choquard; J. Stubbe, The one-dimensional Schrödinger-Newton equations, Lett. Math. Phys., 81, 177-184 (2007) · Zbl 1161.35499 [5] P. Choquard; J. Stubbe; M. Vuffray, Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21, 665-679 (2008) · Zbl 1224.35385 [6] S. Cingolani; M. Clapp; S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6, 891-908 (2013) · Zbl 1260.35198 [7] S. Cingolani; L. Jeanjean, Stationary waves with prescribed \(L^2\)-norm for the Schrödinger-Poisson system, SIAM J. Math. Anal., 51, 3533-3568 (2019) · Zbl 1479.35331 [8] S. Cingolani; S. Secchi; M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140, 973-1009 (2010) · Zbl 1215.35146 [9] S. Cingolani; K. Tanaka, Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35, 1885-1924 (2019) · Zbl 1431.35169 [10] S. Cingolani; T. Weth, On the planar Schrödinger-Poisson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 169-197 (2016) · Zbl 1331.35126 [11] M. Du; T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30, 3492-3515 (2017) · Zbl 1384.35010 [12] R. Harrison; I. Moroz; K. P. Tod, A numerical study of the Schrödinger-Newton equation, Nonlinearity, 16, 101-122 (2003) · Zbl 1040.81554 [13] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2, 1-27 (2009) · Zbl 1183.35266 [14] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math., 57, 93-105 (1977) · Zbl 0369.35022 [15] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072 (1980) · Zbl 0453.47042 [16] S. Masaki, Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space, Comm. Partial Differential Equations, 35, 2253-2278 (2010) · Zbl 1232.35155 [17] V. Moroz; J. Van Schaftingen, Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations, 52, 199-235 (2015) · Zbl 1309.35029 [18] I. M. Moroz; R. Penrose; P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15, 2733-2742 (1998) · Zbl 0936.83037 [19] R. Penrose, On gravity’s role in quantum state reduction, Gen. Rel. Grav., 28, 581-600 (1996) · Zbl 0855.53046 [20] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356, 1927-1939 (1998) · Zbl 1152.81659 [21] R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. · Zbl 1188.00007 [22] J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008. [23] P. Tod; I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12, 201-216 (1999) · Zbl 0942.35077 [24] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp. · Zbl 1189.81061 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.