Existence and multiplicity of sign-changing solutions for quasilinear Schr\"{o}dinger equations with sub-cubic nonlinearity

TL;DR

This paper proves the existence of multiple sign-changing solutions for a class of quasilinear Schrödinger equations with sub-cubic nonlinearities, using a novel perturbation approach and invariant sets method, extending previous results to the case where 2<p<4.

Contribution

It introduces a new perturbation method and invariant sets approach to establish multiple sign-changing solutions without coercivity or monotonicity assumptions.

Findings

Existence of a least energy sign-changing solution.

Infinitely many sign-changing solutions.

Extension to cases with 2<p<4 nonlinearities.

Abstract

In this paper, we consider the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u-u\Delta(u^2)=g(u),\ \ x\in \mathbb{R}^{3}, \end{equation*} where $V$ and $g$ are continuous functions. Without the coercive condition on $V$ or the monotonicity condition on $g$, we show that the problem above has a least energy sign-changing solution and infinitely many sign-changing solutions. Our results especially solve the problem above in the case where $g(u)=|u|^{p-2}u$ ($2<p<4$) and complete some recent related works on sign-changing solutions, in the sense that, in the literature only the case $g(u)=|u|^{p-2}u$ ($p\geq4$) was considered. The main results in the present paper are obtained by a new perturbation approach and the method of invariant sets of descending flow. In addition, in some cases where the functional merely satisfies the Cerami condition, a deformation lemma under the Cerami condition is developed.