On multiplicity and concentration of solutions for a gauged nonlinear Schrödinger equation

Abstract

This paper is concerned with a gauged nonlinear Schrödinger equation $-\mathrm{\Delta }u+\lambda V\left(|x|\right)u+\left(\frac{h^2\left(|x|\right)}{|x{|}^2}+{\int }_{|x|}^{\mathrm{\infty }}\frac{h\left(s\right)}{s}{u}^2\left(s\right)\mathrm{d}s\right)u=f(|x|,u)\text{in}{\mathbb{R}}^2,$ where V is a external potential and λ is a positive parameter. We assume that the potential has a bounded potential well and the nonlinearity is sublinear at infinity, the existence and multiplicity results of solutions are obtained by using the genus properties in critical point theory. Moreover, the concentration behavior of solutions on the potential well is also explored.

COMMUNICATED BY:

• Xingfu Zou

Keywords:

• Gauged Schrödinger equation
• multiplicity
• concentration
• sublinear

2010 Mathematics Subject Classifications:

• 35J20
• 35Q55

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11701173, 11771385, 11601145, 11571370], by the Natural Science Foundation of Hunan Province [grant numbers 2017JJ3130, 2017JJ3131], by the Excellent youth project of Education Department of Hunan Province [grant number 17B143], and by the Hunan University of Commerce Innovation Driven Project for Young Teacher [grant number 16QD008].