Existence and concentrate behavior of positive solutions for Chern–Simons–Schrödinger systems with critical growth

Abstract

In this paper, we investigate the following Chern–Simons–Schrödinger system $-\mathrm{\Delta }u+u+\lambda \left({\int }_{|x|}^{\mathrm{\infty }}\frac{h\left(s\right)}{s}{u}^2\left(s\right)\mathrm{d}s+\frac{h^2\left(|x|\right)}{|x{|}^2}\right)u=f(x,u)\mathrm{i}\mathrm{n}{\mathbb{R}}^2,$ where $\lambda >0$, $h\left(s\right)={\int }_0^s\frac{t}{2}{u}^2\left(t\right)\mathrm{d}t$ and the nonlinearity $f(x,s)\in C({\mathbb{R}}^2\times \mathbb{R},\mathbb{R})$ behaves like $e^{4\pi s^2}$ as $|s|\to +\mathrm{\infty }$. By using the variational methods and the Trudinger–Moser inequality, we obtain the existence of positive solutions for this system. Moreover, we observe the concentrate behavior of mountain-pass type solutions of this system as $\lambda \to 0$.

Keywords:

• Chern–Simons–Schrödinger system
• positive solution
• critical growth
• Trudinger–Moser inequality
• variational method

AMS SUBJECT CLASSIFICATIONS:

• 35B33
• 35B09
• 35J20
• 35D30
• 35A01

Disclosure statement

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

Additional information

Funding

Supported by Postgraduate research and innovation project of Chongqing [No. CYB19082] and National Natural Science Foundation of China [No. 11971393].