The $n$-vortex equation, in the context of optical vortex solitons, is of the form \begin{equation} -(ru_{r})_r+\dfrac{n^2}{r}u+\beta ru= f(u^2)ru,\quad r\in (0,\infty),\\ u(0)=0=u(\infty), \end{equation} where $n$ is a nonzero integer, $\beta$ an undetermined parameter, and $f:\mathbb{R}\rightarrow\mathbb{R}$ can be taken to be a polynomial for simplicity. Are there any uniqueness results for positive solutions of such an equation when $n\neq 0$ on an unbounded domain as given or even on a finite domain $[0,R]$ with $u(0)=0=u(R)$ and $R>0$ sufficiently large? Any comments/references would be highly appreciated.
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