In this paper, we investigate the generalized quasilinear Schrödinger equation \(-\operatorname{div}\left(g^2(u) \nabla u\right)+g(u) g^{\prime}(u)|\nabla u|^2+u=P(\varepsilon x) |u|^{\\\alpha p-2}u, \quad x \in \mathbb{R}^N,\) where N > 3, g: ℝ → ℝ+ is a C1 even function, g(0) = 1, g′(s) ≥ 0 for all s ≥ 0, g(s) = β∣s∣α−1 + O (∣s∣γ−1) as s → ∞ for some constants α ∈ [1, 2], β > 0, γ < α and (α − 1)g(s) ≥ g′(s)s for all s ≥ 0, ε > 0 is a positive parameter, and p ∈ (2, 2*). We will study the impact of the nonlinearity’s coefficient P(x) on the quantity of positive solutions.
