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Solutions of the Schrödinger-Poisson problem concentrating on spheres. II: Existence. (English) Zbl 1187.35236

Author’s abstract: The Schrödinger-Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field. We deal with the semiclassical states for this system and prove the existence of radial solutions concentrating on spheres in the presence of an external potential and with a non-constant density charge. In particular, we show that the necessary conditions obtained in Part I [I. Ianni and G. Vaira, Math. Models Methods Appl. Sci. 19, No. 5, 707–720 (2009; Zbl 1173.35687)] are also sufficient if suitable non-degeneracy conditions are assumed. We use a perturbation technique in a variational setting.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35A15 Variational methods applied to PDEs

Citations:

Zbl 1173.35687
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References:

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[2] DOI: 10.1007/s00220-003-0811-y · Zbl 1072.35019
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[10] DOI: 10.1002/cpa.3160390103 · Zbl 0594.35005
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