Multiplicity and Concentration Results for Fractional Schrödinger-Poisson Equations with Magnetic Fields and Critical Growth

We deal with the following fractional Schrödinger-Poisson equation with magnetic field ε2s(−Δ)A/εsu+V(x)u+ε−2t(|x|2t−3∗|u|2)u=f(|u|2)u+|u|2s∗−2uin ℝ3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon^{2s}(-{\Delta})_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{{2}_{s}^{*}-2}u \quad \text{ in } \mathbb{R}^{3}, $$\end{document} where ε > 0 is a small parameter, 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}, t ∈ (0, 1), 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 exponent, (−Δ)As\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-{\Delta })^{s}_{A}$\end{document} is the fractional magnetic Laplacian, V:ℝ3→ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V:\mathbb {R}^{3}\rightarrow \mathbb {R}$\end{document} is a positive continuous potential, A:ℝ3→ℝ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A:\mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$\end{document} is a smooth magnetic potential and f:ℝ→ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:\mathbb {R}\rightarrow \mathbb {R}$\end{document} is a subcritical nonlinearity. Under a local condition on the potential V, we study the multiplicity and concentration of nontrivial solutions as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon \rightarrow 0$\end{document}. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential V attains its minimum.