Sign-changing solutions for a modified nonlinear Schrödinger equation in $${\mathbb {R}}^N$$ R N

We consider the modified nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+V(x)u-\Delta (u^2)u =|u|^{p-2}u\ \ \text{ in }\ {\mathbb {R}}^N, \end{aligned}$$ where $$N\ge 3,\ V$$ is a given potential and $$p>2$$ . The equation appears in modeling of superfluid film theory in plasma physics. While most existing works in the literature are only for $$p\in [4,\,4N/(N-2))$$ , we propose a new variational approach to deal with exponents $$p\in (2,\,4N/(N-2))$$ in a unified way and obtain infinitely many sign-changing solutions.