Remarks on Catalan's equation over function fields

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