Dengfeng Lü, Shuwei Dai, A remark on Chern-Simons-Schrödinger equations with Hartree type nonlinearity

Full Text: PDF
DOI: 10.23952/jnva.7.2023.3.06

Volume 7, Issue 3, 1 June 2023, Pages 409-420

Abstract. This paper is devoted to studying the following Chern-Simons-Schrödinger equation with Hartree type nonlinearity:

$-\frac{1}{2m}\Delta \psi+\omega \psi+\frac{2e^{4}}{m\kappa^{2}}\left( \int_{|x|}^{+\infty}\frac{a(\tau)}{\tau}\psi^{2}(\tau)\mathrm{d}\tau +\frac{a^{2}(|x|)}{|x|^{2}}\right)\psi=(R_{\alpha}\ast F(\psi))F'(\psi)$ in $\mathbb{R}^{2},$

where $e>0$ is a parameter, $m,\omega,\kappa>0$ are constants, $a(\tau)=\frac{1}{2}\int_{0}^{\tau}s \psi^{2}(s)\mathrm{d}s$ , and $F\in C^{1}(\mathbb{R},\mathbb{R})$ . By using variational methods and perturbation arguments, the existence of positive solutions for the above equation is derived. In addition, the asymptotic behavior of solutions with regard to the parameter $e$ is also considered.

How to Cite this Article:
D. Lü, S. Dai, A remark on Chern-Simons-Schrödinger equations with Hartree type nonlinearity, J. Nonlinear Var. Anal. 7 (2023), 409-420.