Nodal Solutions for Quasilinear Schrödinger Equations with Asymptotically 3-Linear Nonlinearity

Abstract

In this paper, we are concerned with the quasilinear Schrödinger equation

$$\begin{aligned} -\Delta u+V(x)u-u\Delta (u^2)=g(u),\ \ x\in \mathbb {R}^{N}, \end{aligned}$$

where $N\ge 3$, V is radially symmetric and nonnegative, and g is asymptotically 3-linear at infinity. In the case of $\inf _{\mathbb {R}^N}V>0$, we show the existence of a least energy sign-changing solution with exactly one node, and for any integer $k>0$, there is a pair of sign-changing solutions with k nodes. Moreover, in the case of $\inf _{\mathbb {R}^N}V=0$, the problem above admits a least energy sign-changing solution with exactly one node. The proof is based on variational methods. In particular, some new tricks and the method of sign-changing Nehari manifold depending on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically 3-linear nonlinearities.