  In this paper, we are concerned with semiclassical states to the following
fractional nonlinear elliptic equation, \begin{align*} \eps^{2s}(-\Delta)^s u +
V(x) u=\mathcal{N}(|u|)u \quad \mbox{in} \,\,\, \R^N, \end{align*} where $0<s
<1$, $\eps>0$ is a small parameter, $N>2s$, $V \in C^1(\R^N, \R^+)$ and
$\mathcal{N}\in C(\R, \R)$. The nonlinearity has Sobolev subcritical, critical
or supercritical growth. The fractional Laplacian $(-\Delta)^s$ is
characterized as $\mathcal{F}((-\Delta)^{s}u)(\xi)=|\xi|^{2s}
\mathcal{F}(u)(\xi)$ for $\xi \in \R^N$, where $\mathcal{F}$ denotes the
Fourier transform. We construct positive semiclassical states and an infinite
sequence of sign-changing semiclassical states with higher energies clustering
near the local minimum points of the potential $V$. The solutions are of higher
topological type, which are obtained from a minimax characterization of higher
dimensional symmetric linking structure via the symmetric mountain pass
theorem. They correspond to critical points of the underlying energy functional
at energy levels where compactness condition breaks down. The proofs are mainly
based on penalization methods, s-harmonic extension theories and blow-up
arguments along with local type Pohozaev identities.
