A multiplicity result for Chern-Simons-Schr\"odinger equation with a general nonlinearity

TL;DR

This paper establishes the existence of multiple solutions for a Chern-Simons-Schrödinger equation with a general nonlinearity in , using variational methods and under broad assumptions on the nonlinear term.

Contribution

It provides a new multiplicity result for solutions of the Chern-Simons-Schrödinger equation with general nonlinearities, extending previous results to more general settings.

Findings

At least n solutions exist for each n in  under certain conditions.

Solutions exist for a range of the parameter q in (0, q_n).

The results apply to broad classes of nonlinearities g.

Abstract

In this paper we give a multiplicity result for the following Chern-Simons-Schr\"odinger equation \[ -\Delta u+2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds +q u\frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad\hbox{in }\mathbb{R}^2, \] where $\displaystyle h_u(s)=\int_0^s \tau u^2(\tau) \ d \tau$, under very general assumptions on the nonlinearity $g$. In particular, for every $n\in \mathbb N$, we prove the existence of (at least) $n$ distinct solutions, for every $q\in (0,q_{n})$, for a suitable $q_n$.