We deal with the following fractional Schr\"odinger-Poisson equation with magnetic field \begin{ equation} \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 \mbox{ in} \mathbb{ R}^{ 3}, \nonumber \end{ equation} where $\varepsilon>0$ is a small parameter, $s\in (\frac{ 3}{ 4}, 1)$, $t\in (0,1)$, $2^{ *}_{ s}=\frac{ 6}{ 3-2s}$ is the fractional critical exponent, $(-\Delta)^{ s}_{ A}$ is the fractional magnetic Laplacian, $V:\mathbb{ R}^{ 3}\rightarrow \mathbb{ R}$ is a positive continuous potential, $A:\mathbb{ R}^{ 3}\rightarrow \mathbb{ R}^{ 3}$ is a smooth magnetic potential and $f:\mathbb{ R}\rightarrow \mathbb{ R}$ is a subcritical nonlinearity. Under a local condition on the potential $V$, we study multiplicity and concentration of nontrivial solutions as $\varepsilon\rightarrow 0$. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential $V$ attains its minimum.
Categories: math.AP