Remarks on Catalan's equation over function fields

Let $\mathrm{\ell }$ be a prime number, $F$ be a global function field of characteristic $\mathrm{\ell }$. Assume that there is a prime $P_{\mathrm{\infty }}$ of degree $1$. Let ${\mathcal{O}}_F$ be the ring of functions in $F$ with no poles outside of $\{P_{\mathrm{\infty }}\}$. We study solutions to Catalan’s equation $X^m-Y^n=1$ over ${\mathcal{O}}_F$ and show that under certain additional conditions, there are no non-constant solutions which lie in ${\mathcal{O}}_F$, when $m,n>1$.