Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials

This paper deals with the following fractional magnetic Schrödinger equations

$ \varepsilon^{2s}(-\Delta)^s_{A/\varepsilon} u +V(x)u = |u|^{p-2}u, \ x\in{\mathbb R}^N, $

where $ \varepsilon>0 $ is a parameter, $ s\in(0,1) $, $ N\geq3 $, $ 2+2s/(N-2s)<p<2_s^*: = 2N/(N-2s) $, $ A\in C^{0,\alpha}({\mathbb R}^N,{\mathbb R}^N) $ with $ \alpha\in(0,1] $ is a magnetic field, $ V:{\mathbb R}^N\to{\mathbb R} $ is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of $ V $ as $ \varepsilon\to 0 $. There is no restriction on the decay rates of $ V $. Especially, $ V $ can be compactly supported. The appearance of $ A $ and the nonlocal of $ (-\Delta)^s $ makes the proof more difficult than that in [7], which considered the case $ A\equiv 0 $.