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Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

Authors
  • Yang, Zhipeng1, 2
  • Zhao, Fukun1
  • 1 Department of Mathematics, Yunnan Normal University, P.R.China , (China)
  • 2 Mathematical Institute, Georg-August-University of Göttingen, Germany , (Germany)
Type
Published Article
Journal
Advances in Nonlinear Analysis
Publisher
De Gruyter
Publication Date
Dec 01, 2020
Volume
10
Issue
1
Pages
732–774
Identifiers
DOI: 10.1515/anona-2020-0151
Source
De Gruyter
Keywords
License
Green

Abstract

In this paper, we study the singularly perturbed fractional Choquard equation ε2s(−Δ)su+V(x)u=εμ−3(∫R3|u(y)|2μ,s∗+F(u(y))|x−y|μdy)(|u|2μ,s∗−2u+12μ,s∗f(u))inR3, $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, 2μ,s⋆=6−μ3−2s $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

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