Semiclassical states for fractional logarithmic Schr\"{o}dinger equations

In this paper, we consider the following fractional logarithmic Schr"odinger equation \begin{equation*} \varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log |u|^2\
$\text{in}$\ \R^N, \end{equation*} where $\epsilon >0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\epsilon }$ concentrating at a local minimum of $V$ as $\epsilon \to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.