  In this paper we study the multiplicity and concentration of positive
solutions for the following $(p, q)$-Laplacian problem: \begin{equation*}
\left\{ \begin{array}{ll} -\Delta_{p} u -\Delta_{q} u +V(\varepsilon x)
\left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\
u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in
} \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is
a small parameter, $1< p<q<N$, $\Delta_{r}u=\mbox{div}(|\nabla u|^{r-2}\nabla
u)$, with $r\in \{p, q\}$, is the $r$-Laplacian operator,
$V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous function satisfying
the global Rabinowitz condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a
continuous function with subcritical growth. Using suitable variational
arguments and Ljusternik-Schnirelmann category theory, we investigate the
relation between the number of positive solutions and the topology of the set
where $V$ attains its minimum for small $\varepsilon$.
