SEMICLASSICAL SOLUTIONS OF THE CHOQUARD EQUATIONS IN $\mathbb{R}$<sup>3</sup>

Ke Jin; Zifei Shen

doi:10.11948/20200388

2021 Volume 11 Issue 1

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Ke Jin, Zifei Shen. SEMICLASSICAL SOLUTIONS OF THE CHOQUARD EQUATIONS IN $\mathbb{R}$3[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 568-586. doi: 10.11948/20200388

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Ke Jin, Zifei Shen. SEMICLASSICAL SOLUTIONS OF THE CHOQUARD EQUATIONS IN $\mathbb{R}$³[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 568-586. doi: 10.11948/20200388

SEMICLASSICAL SOLUTIONS OF THE CHOQUARD EQUATIONS IN $\mathbb{R}$³

Ke Jin¹,
Zifei Shen^1, ,

1.
Department of Mathematics, Zhejiang Normal University, 321004, Jinhua, Zhejiang, China

Corresponding author: Email: szf@zjnu.edu.cn(Z. Shen)
Fund Project: Zifei Shen is supported by National Natural Science Foundation of China (11671364) and National Natural Science Foundation of China (12071438)

Abstract

Abstract

We study the nonlocal equation:

$ - {\varepsilon ^2}\Delta u + \lambda u + V(x)u = {\varepsilon ^{ - 2}}\left( {|x{|^{ - 1}} * |u{|^p}} \right)|u{|^{p - 2}}u\quad {\rm{ in }}\ {\mathbb{R}^3}, $

where ε > 0 is a small parameter, λ > 0, 0 < p < ∞ are positive constants and u is a real-valued measurable function. By Lyapunov–Schmidt reduction, we will prove the existence of multiple semiclassical solutions.
- Choquard equations /
- semiclassical states /
- Lyapunov–Schmidt reduction
MSC: 35J20, 35J60

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References

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