Semiclassical solutions for fractional logarithmic Schrödinger equations with potentials unbounded below

In this paper, we consider the following fractional logarithmic Schrödinger equation \begin{equation*} \varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log u^2\quad \text{in } \R^N, \end{equation*} where $\varepsilon> 0$, $N\ge 1$ and $V(x)\in C\big(\R^N,\R\big)$ is a potential which can be unbounded below at infinity. By considering a new penalization, we show that the problem has a nontrivial solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$.