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

Abstract

<p style='text-indent:20px;'>This paper deals with the following fractional magnetic Schrödinger equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \varepsilon^{2s}(-\Delta)^s_{A/\varepsilon} u +V(x)u = |u|^{p-2}u, \ x\in{\mathbb R}^N, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ N\geq3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 2+2s/(N-2s)&lt;p&lt;2_s^*: = 2N/(N-2s) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ A\in C^{0,\alpha}({\mathbb R}^N,{\mathbb R}^N) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \alpha\in(0,1] $\end{document}</tex-math></inline-formula> is a magnetic field, <inline-formula><tex-math id="M7">\begin{document}$ V:{\mathbb R}^N\to{\mathbb R} $\end{document}</tex-math></inline-formula> is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of <inline-formula><tex-math id="M8">\begin{document}$ V $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon\to 0 $\end{document}</tex-math></inline-formula>. There is no restriction on the decay rates of <inline-formula><tex-math id="M10">\begin{document}$ V $\end{document}</tex-math></inline-formula>. Especially, <inline-formula><tex-math id="M11">\begin{document}$ V $\end{document}</tex-math></inline-formula> can be compactly supported. The appearance of <inline-formula><tex-math id="M12">\begin{document}$ A $\end{document}</tex-math></inline-formula> and the nonlocal of <inline-formula><tex-math id="M13">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> makes the proof more difficult than that in [<xref ref-type="bibr" rid="b7">7</xref>], which considered the case <inline-formula><tex-math id="M14">\begin{document}$ A\equiv 0 $\end{document}</tex-math></inline-formula>.</p>