\begin{equation}\tag{$P_\varepsilon$}

\left\{ \begin{array}{ll}

    -\varepsilon^{p} \Delta_p u + V(x)u^{p-1} = h(u), \ \text{ em \:} \R^N, \\

    u > 0  \text{\: em \:} \mathbb{R}^N, \ u \in W^{1,p} (\mathbb{R}^N),

\end{array}

\right.

\end{equation}

where $\varepsilon > 0$ is a positive parameter, $2 \leq p<N $, $\Delta_p u = div(|\Grad u|^{p-2} \Grad u)$ denotes the p-Laplacian operator, $h: \R \to \R$ is a continuous function satisfying some conditions and $V: \R \to \R$ is a function of class $C^2$ which belongs to two classes of potentials. Our study is divided in two parts: firstly we show the same results obtained by (Alves, 2015) for $p \geq 2$, establishing the existence of positive solution for the \eqref{P} problem when $h$ has subcritical growth; in the second we show the existence of positive solution considering $h$ with critical growth. The main tools used are the Variational Methods, Mountain Pass Theorem, Lions' Concentration-Compactness Principle and del Pino and Felmer's Penalization Method.