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

Abstract

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 $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.