  In this work we consider the following class of fractional $p\&q$ Laplacian
problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u +
V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N},
\end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1<
p<q<\frac{N}{s}$, $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional
$t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a
continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a
$\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems
and the Ljusternik-Schnirelmann theory, we investigate the existence,
multiplicity and concentration of nontrivial solutions provided that
$\varepsilon$ is sufficiently small.
