Abstract: 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 $N>3$, $g\!\!:\mathbb{R} \rightarrow \mathbb{R}^{+}$ is a $C^1$ even function, $g(0)=1$, $g^{\prime}(s) \geq 0$ for all $s \geq 0$, $g(s)=\beta|s|^{\alpha-1}+O\left(|s|^{\gamma-1}\right)$ as $s \rightarrow \infty$ for some constants $\alpha \in[1,2]$, $\beta>0$, $\gamma<\alpha$ and $(\alpha-1) g(s) \geq g^{\prime}(s) s$ for all $s \geq 0$, $\varepsilon>0$ is a positive parameter, and $p \in\left(2,2^*\right)$. We will study the impact of the nonlinearity's coefficient $P(x)$ on the quantity of positive solutions.

Key words: variational methods, generalized quasilinear equation