  We revisit the following fractional Schr\"{o}dinger equation
  \begin{align}\label{1a}
  \varepsilon^{2s}(-\Delta)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N,
  \end{align} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes
the fractional Laplacian, $s\in(0,1)$, $p\in (2, 2_s^*)$, $2_s^*=\frac
{2N}{N-2s}$, $N>2s$, $V\in C\big(\R^N, [0, +\infty)\big)$ is a potential. Under
various decay assumptions on $V$, we introduce a uniform penalization argument
combined with a comparison principle and iteration process to detect an
explicit threshold value $p_*$, such that the above problem admits positive
concentration solutions if $p\in (p_*, \,2_s^*)$, while it has no positive weak
solutions for $p\in (2,\,p_*)$ if $p_*>2$, where the threshold $p_*\in [2,
2^*_s)$ can be characterized explicitly by \begin{equation*}\label{qdj111}
p_*=\left\{\begin{array}{l}
  2+\frac {2s}{N-2s} \ \ \ \text { if } \lim\limits_{|x| \to \infty}
(1+|x|^{2s})V(x)=0,\vspace{1mm}
  2+\frac {\omega}{N+2s-\omega} \text { if } 0<\inf (1+|x|^\omega)V(x)\le \sup
(1+|x|^\omega)V(x)< \infty \text { for some } \omega \in [0, 2s],\vspace{1mm}
  2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { if } \inf V(x)\log(e+|x|^2)>0.
\end{array}\right. \end{equation*} Moreover, corresponding to the various decay
assumptions of $V(x)$, we obtain the decay properties of the solutions at
infinity.
