The Existence of Arbitrary Multiple Nodal Solutions for a Class of Quasilinear Schrödinger Equations

This paper is concerned to studying the quasilinear Schrödinger equation: \(\begin{aligned} -\Delta u+V(x)u-\frac{\gamma }{2}\Delta (u^2)u=|u|^{p-2}u,~~x\in \mathbb {R}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+V(x)u-\frac{\gamma }{2}\mathrm{\Delta }(u^2)u={|u|}^{p-2}u,x\in {\mathbb{R}}^N,\end{array}$$ where V(x) is a given potential, \(\gamma >0\) $\gamma >0$ and either \(p\in (2,2^*),2^*=\frac{2N}{N-2}\) $p\in (2,2^{\ast }),2^{\ast }=\frac{2N}{N-2}$ for \(N\geqslant 4\) $N⩾4$ or \(p\in (2,4)\) $p\in (2,4)$ for \(N=3\) $N=3$ . We establish the existence of arbitrary multiple nodal solutions for the above equations.