On the number of concentrating solutions of a fractional Schrödinger–Poisson system with doubly critical growth

In this paper we study the existence, multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson system with doubly critical growth ε2s(-Δ)su+V(x)u=f(u)+ϕ|u|2s∗-3u+|u|2s∗-2u,x∈R3,ε2s(-Δ)sϕ=|u|2s∗-1,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll}\displaystyle \varepsilon ^{2s}(-\Delta )^{s}u+V( x)u= f(u)+\phi |u|^{2^*_s-3}u+|u|^{2^{*}_{s}-2}u, &{} \quad x \in {\mathbb {R}}^{3},\\ \varepsilon ^{2s}(-\Delta )^{s}\phi =|u|^{2^*_s-1}, &{} \quad x \in {\mathbb {R}}^{3},\\ \end{array}\right. } \end{aligned}$$\end{document}where s∈(34,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (\frac{3}{4},1)$$\end{document}, 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, 2s∗=63-2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^*_s = \frac{6}{3-2s}$$\end{document} is the fractional critical Sobolev exponent, (-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} is the fractional Laplacian operator, and f is a continuous nonlinearity with subcritical growth. With the help of Nehari manifold and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value for small values of the parameter 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}.