We are concerned with the concentration behavior of semi-classical states of the following N-Laplacian Schrödinger equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{N}\Delta _N v+V(x)|v|^{N-2}v=f(v)\ \ \text{ in } \ \ \mathbb {R}^{N},\\ v>0,\ v(x)\rightarrow 0 \ \text {as}\ |x|\rightarrow +\infty .\;\;& \end{array}\right. } \end{aligned}\) Here \(N\ge 2\) , \(\varepsilon >0\) is a small parameter, \(\Delta _N\) is the N-Laplacian operator, i.e., \(\Delta _N u:=\text {div}(|\nabla u|^{N-2}\nabla u)\) . The potential V has a local maximum point, and f is assumed to be of critical growth in the sense of the Trudinger–Moser inequality. We define a penalization related to the barycenters of functions which is introduced by Zhang and Zhang (2023 Nonlinearity 36 3125–3157) and apply the Brouwer degree theory to prove the existence of semi-classical states for the N-Laplacian Schrödinger equation. As \(\varepsilon \rightarrow 0\) , these semi-classical states concentrate around the maximum point of the potential function V. Here we adopt the local variational methods from Byeon and Jeanjean (2007 Arch. Ration. Mech. Anal. 185 185–200); Byeon and Tanaka (2013 J. Eur. Math. Soc. 15 1859–99; 2014 Mem. Amer. Math. Soc. 229 viii+89 pp).
