In this work we are interested in studying the existence of a solution and the phenomenon of solution concentration for the following class of problems \(\begin{aligned} -\operatorname {div} (\hbar ^2 \phi (\hbar |\nabla u |)\nabla u) + V(x)b(|u|)u + W'(u) = 0, \quad x \in \mathbb {R}^N, \quad (P_\hbar ) \end{aligned}\) where \( \hbar > 0 \) , \( u: \mathbb {R}^N \rightarrow \mathbb {R}^{N+1}\) , \( 3 \le N < p \) , \( \Phi \) is a N-function of the form \(\begin{aligned} \Phi (t) = \int _{0}^{|t|} s\phi (s) ds, \end{aligned}\) the function \( b:(0,+\infty ) \rightarrow (0,+\infty ) \) checks the relationship \( B(t) = \int _{0}^{|t|} sb(s) ds, \) where B is a N-function, and V, W satisfy some technical conditions.
