Existence and multiplicity of sign-changing solitary waves for a quasilinear Schrödinger equation

ABSTRACT

This paper is motivated by the study of the following quasilinear Schrödinger equation $-\mathrm{\Delta }u+V\left(x\right)u-\left[\mathrm{\Delta }\right(1+u^2{)}^{\frac{1}{2}}]\frac{u}{2(1+u^2{)}^{\frac{1}{2}}}=\lambda h(u),x\in {\mathbb{R}}^N,$where $N\ge 3$, $\lambda >0$ is a parameter and $V\left(x\right)$ is a given positive potential. As an example, the nonlinearity includes the pure power type of $h\left(u\right)=|u{|}^{p-2}u$ for the well-studied case $12-4\sqrt{6}<p<2^{\ast }$, and the case $2<p<12-4\sqrt{6}$ in which few existence results are known. Distinguishing from the existing results in the literature, we are more interested in the existence and multiplicity of sign-changing solutions for the above problem.

Keywords:

• Quasilinear Schrödinger equation
• sign-changing solutions
• invariant sets of descending flow

MSC 2010:

• 35J20
• 35J62
• 58E05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

W. Huang was supported by the Natural Science Foundation of China [grant number 12001198] and the Natural Science Foundation of Jiangxi Province [grant number 20202BABL211004]. L. Wang was supported by the Natural Science Foundation of China [grant number 11701178]. Q. Wang was supported by the Natural Science Foundation of China [grant number 11701439].