Summary: In this paper we study the quasilinear problem \[ \begin{cases} &-\operatorname{div}(a(|\nabla u|^p)|\nabla u|^{p-2}\nabla u)+[1+ \mu V(z)]b(|u|^p) |u|^{p-2}u=f(u)+\varrho|u|^{\sigma-2}u,\\ &\qquad u \in W^{1,p}(\mathbb{R}^N)\cap W^{1,q}(\mathbb{R}^N). \end{cases} \] The term \(1+\mu V(z)\) is the steep potential well introduced by T. Bartsch and Z.-Q. Wang in [Commun. Partial Differ. Equations 20, No. 9–10, 1725–1741 (1995; Zbl 0837.35043)]. With suitable hypotheses on the functions \(a\), \(b\) and \(f\), we show the existence of solutions and concentration behavior occurred as \(\mu\to+\infty\), considering the subcritical case, the critical case and the supercritical case.