Sign-changing solitary waves for a quasilinear Schrödinger equation with general nonlinearity

Abstract

This research is undertaken with the primary objective of exploring a quasilinear Schrödinger equation, a mathematical model of significant importance in describing diverse physical phenomena. Specifically, we direct our focus to the following equation: $-\mathrm{\Delta }u+V(x)u-[\mathrm{\Delta }(1+u^2{)}^{1/2}]\frac{u}{2(1+u^2{)}^{\frac{1}{2}}}=h(u),x\in {\mathbb{R}}^N,$ where $N\ge 3$, V is a given positive potential and h represents a general nonlinearity. Employing an innovative perturbation technique and the method of invariant sets in the descending flow, we rigorously establish the existence and multiplicity of sign-changing solutions for the aforementioned problem. In particular, for pure power type nonlinearity $h(u)=|u{|}^{p-2}u$, we are concerned mostly with $2<p\le 12-4\sqrt{6}$.

KEYWORDS:

• Quasilinear Schrödinger equation
• sign-changing solutions
• perturbation technique
• invariant sets in the descending flow

Acknowledgments

The authors would like to thank the anonymous referees for carefully reading this paper and making valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Wentao Huang was supported by the Natural Science Foundation of China [grant number 12001198] and the Natural Science Foundation of Jiangxi Province [grant number 20232BAB201009]. Li Wang was supported by the Natural Science Foundation of China [grant number 12161038].