Multiple normalized solutions of Chern–Simons–Schrödinger system

Abstract

In this paper, we consider the following equation $$-\Delta u + \omega u + \left(\int_{|x|}^{\infty}\frac{h(s)}{s}u^2(s)ds\right) u + \frac{h^2(|x|)}{|x|^2}u - \lambda|u|^{p - 2}u = 0\quad \mbox{in}\quad \mathbb{R}^2,$$ for p > 2 and $${\lambda > 0}$$ , which appeared in Byeon et al. (J Funct Anal 263(6):1575–1608, 2012) to find the standing wave solutions of the Chern–Simons–Schrödinger system. By using the minimax theorem, we get the multiplicity results for the L 2-normalized solutions to the equation, and thus there are multiple L 2-normalized solutions of the Chern–Simons–Schrödinger system.