Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 285-302

MULTIPLE POSITIVE SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER–POISSON SYSTEM

Weiming Liu and Miaomiao Niu

Hubei Normal University, School of Mathematics and Statistics
Huangshi, 435002, P.R. China; whu.027 'at' 163.com

Beijing Normal University, School of Mathematical Sciences
Beijing, 100875, P.R. China; miaomiaoniu 'at' mail.bnu.edu.cn

Abstract. We consider the following Schrödinger–Poisson system in R³

(0.1) -Δu + u + αK(|x|)Φ(x)u =|u|^p-2u, x ∈ R³,
-ΔΦ = K(|x|)u², x ∈ R³,

where 2 < p < 6, α can be regarded as a parameter and K(r) (r = |x|) is a positive continuous function. There are constants a ∈ R and b ∈ (0,1/2], such that K(r) ∼ r^ae^-br, as r → +∞. Then, (0.1) possesses a non-radial positive solution with exactly m maximum points for suitable range of α.

2010 Mathematics Subject Classification: Primary 35J10, 35J20, 35J60.

Key words: Schrödinger–Poisson system, non-radial positive solutions, variational methods, Lyapunov–Schmidt reduction.

Reference to this article: W. Liu and M. Niu: Multiple positive solutions for the nonlinear Schrödinger–Poisson system. Ann. Acad. Sci. Fenn. Math. 42 (2017), 285-302.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4218