Nodal solutions for quasilinear Schr\"{o}dinger equations with asymptotically 3-linear nonlinearity

TL;DR

This paper establishes the existence of nodal solutions for a class of quasilinear Schrödinger equations with asymptotically 3-linear nonlinearities using variational methods and novel techniques involving sign-changing Nehari manifolds.

Contribution

It introduces new variational techniques and a sign-changing Nehari manifold approach to handle asymptotically 3-linear nonlinearities in Schrödinger equations.

Findings

Existence of least energy sign-changing solutions with one node.

Existence of multiple sign-changing solutions with arbitrary nodes.

Applicable to cases where the potential V is positive or zero.

Abstract

In this paper, we are concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u-u\Delta(u^2)=g(u),\ \ x\in \mathbb{R}^{N}, \end{equation*} where $N\geq3$, $V$ 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 are 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.