Normalized solutions for (p,N)-Laplacian equations with exponential critical growth and competing potentials: Existence, multiplicity and concentration

This paper studies the existence of normalized solutions for the [Formula: see text]-Laplacian equation with competing potentials and exponential critical growth: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] represent the competing absorption and reaction potentials respectively, [Formula: see text] is the Lagrange multiplier, and the nonlinearity [Formula: see text] exhibits exponential critical growth. While normalized solutions with competing potentials remain largely unexplored, especially in the context of nonstandard growth operators, we provide the first systematic investigation in this setting. By adopting truncation techniques combined with variational methods, we establish the existence of normalized solutions that concentrate at specific locations for small [Formula: see text]. Our analysis reveals that the concentration points are determined by the balanced competition between the absorption and reaction potentials. Furthermore, using Lusternik–Schnirelmann category theory, we obtain multiplicity results relating the number of solutions to the topological structure of the potentials’ extremal sets. This work provides new insights into the concentration phenomena of normalized solutions under potential competition.