Multiplicity of semiclassical solutions for fractional Choquard equations with critical growth

In this article, we focus on the following fractional Choquard equation involving upper critical exponent ε2s(-Δ)su+V(x)u=εμ-N[|x|-μ∗|u|2μ,s∗]|u|2μ,s∗-2u+λf(u),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon ^{2s}(-\Delta )^su+V(x)u=\varepsilon ^{\mu -N}[|x|^{-\mu }*|u|^{2_{\mu ,s}^*}]|u|^{2_{\mu ,s}^*-2}u+\lambda f(u), \ x \in \mathbb {R}^N,\quad \end{aligned}$$\end{document}where ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is a positive parameter, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document}, 0<s<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<1$$\end{document}, (-Δ)s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document} denotes the fractional Laplacian of order s, N>2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>2s$$\end{document}, 0<μ<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\mu <N$$\end{document} and 2μ,s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2_{\mu ,s}^*$$\end{document} is fractional critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Under suitable assumptions on the potential V and nonlinearity f, using variational tools from Nehari manifold method and Ljusternik–Schnirelmann category theory, we establish the existence and multiplicity of semiclassical positive solutions.