Standing wave solutions for a generalized quasilinear Schrödinger equation with indefinite potential

Abstract

This paper is concerned with the standing waves for a generalized quasilinear Schrödinger equation with indefinite potential. By Morse theory, we obtain a nontrivial solution for the equation. In addition, if the nonlinear term is an odd function, we can obtain an unbounded sequence of solutions.

Keywords:

• Schrödinger equation
• quasilinear equation
• standing waves
• Morse theory

2010 Mathematics Subject Classifications:

• 35J20
• 35J60
• 35J62

Acknowledgments

The author would like to thank the handling editors and anonymous referee for the help in the processing of the paper.

Disclosure statement

The author declare that they have no competing interest(s).

Authors' contributions

In the paper, we established the existence of nontrivial solutions for problem (2(2) $-\mathrm{\Delta }u+V(x)u-\mathrm{\Delta }l(u^2)l^{\prime }(u^2)u=h(u),x\in {\mathbb{R}}^N,$(2) ) with the general function $l(s)$ and the indefinite potential.

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

This work was supported by NSFC, China [grant numbers 11771170, 11701203] and Research Start-up Fund of Jianghan University [grant number 06050001].