  This article focuses on the study of the existence, multiplicity and
concentration behavior of ground states as well as the qualitative aspects of
positive solutions for a $(p, N)$-Laplace Schr\"{o}dinger equation with
logarithmic nonlinearity and critical exponential nonlinearity in the sense of
Trudinger-Moser in the whole Euclidean space $\mathbb{R}^N$. Through the use of
smooth variational methods, penalization techniques, and the application of the
Lusternik-Schnirelmann category theory, we establish a connection between the
number of positive solutions and the topological properties of the set in which
the potential function achieves its minimum values.
