In this paper, we consider the modified Schrödinger equation \(\begin{aligned} -\Delta u+V(x) u-\frac{\gamma }{2} u\Delta (u^{2})=|u|^{p-2}u,\ x\in {\mathbb {R}}^{N}, \end{aligned}\) where \(V: {{\mathbb {R}}}^N\rightarrow {{\mathbb {R}}}\) is a continuous potential allowed to change sign and \(\gamma >0\) is a real parameter. This equation is related to standing wave solutions of a class of quasilinear Schrödinger equations which is used for describing the superfluid film equations in plasma physics and has attracted much attention in recent years. It follows from recent many studies that \(p\ge 4\) plays a key role to apply mountain pass theorem. By truncation technique and variational methods, our study establish the existence of multiple solutions for \(p\in (2,\frac{2N}{N-2})\) with \(N\ge 4\) and \(p\in (2,4)\) with \(N=3\) .
