Multiple solutions to a magnetic nonlinear Choquard equation

Cingolani, Silvia; Clapp, Mónica; Secchi, Simone

doi:10.1007/s00033-011-0166-8

Cited by 198 publications

(111 citation statements)

References 31 publications

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“…Ma and Zhao considered the generalized Choquard Equation for q ≥2, and they proved that every positive solution of is radially symmetric and monotone decreasing about some point, under the assumption that a certain set of real numbers, defined in terms of N , α , and q , is nonempty. Under the same assumption, Cingolani, Clapp, and Secchi gave some existence and multiplicity results in the electromagnetic case and established the regularity and some decay asymptotically at infinity of the ground states of . In , Moroz and Van Schaftingen eliminated this restriction and showed the regularity, positivity, and radial symmetry of the ground states for the optimal range of parameters and derived decay asymptotically at infinity for them as well.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Ground states for nonlinear fractional Choquard equations with general nonlinearities

Shen

Gao

Yang

2016

Math Methods in App Sciences

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Section: Introduction and Main Resultsmentioning

confidence: 93%

Ground states for nonlinear fractional Choquard equations with general nonlinearities

Shen

Gao

Yang

2016

Math Methods in App Sciences

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“…In [28] Penrose derived (1.3) in his discussion about the self gravitational collapse of a quantum-mechanical system. Recently, existence and regularity results have also been obtained for a(x) ≡ λ and for more general convolution potentials, see [1,13,18,23,25].…”

Section: Introductionmentioning

confidence: 97%

On the planar Schrödinger–Poisson system

Cingolani

Weth

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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165

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We develop a variational framework to detect high energy solutions of the planar Schrödingerwith a positive function a ∈ L ∞ (R 2 ) and γ > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z 2 -translations and therefore fails to satisfy a global Palais-Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u, w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u > 0 in R 2 and w(x) → −∞ as |x| → ∞ are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation.

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“…(1) has also spurred a great deal of activity in the mathematical community. See, e.g., [6,12,14,15,18,24,25,[34][35][36]38,47,51]. …”

Section: Introductionmentioning

confidence: 99%

Initial condition dependence and wave function confinement in the Schrödinger–Newton equation

et al. 2015

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In this work we study the dynamics of the Schrödinger-Newton (SN) equation upon different choices of initial conditions. Setting up superpositions of Gaussian-like wave packages, a very rich behavior for the critical mass as a function of the parameters of the problem is observed. We find that, for certain values of the parameters, the critical mass is smaller than the critical mass for the system whose initial condition is a single Gaussian wave package, which was the situation previously investigated in the literature. This opens a possibility that more complex initial conditions could in fact produce a significant decrease in the value of the critical mass, which could imply that the SN approach could be tested experimentally. Our conclusions rely on both numerical and analytic estimates. Furthermore, a detailed numerical study is carried out in order to investigate finite-size effects on the simulations, refining earlier results already published. In order to facilitate the reproducibility of our results, a detailed description of our numerical methods has been included in the presentation. Electronic supplementary materialThe online version of this article (

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Multiple solutions to a magnetic nonlinear Choquard equation

Cited by 198 publications

References 31 publications

Ground states for nonlinear fractional Choquard equations with general nonlinearities

Ground states for nonlinear fractional Choquard equations with general nonlinearities

On the planar Schrödinger–Poisson system

Initial condition dependence and wave function confinement in the Schrödinger–Newton equation

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