Non-trivial Solutions of $A{a}^p+B{b}^p=C{c}^3$ over Number Fields

In this paper, we investigate solutions to the Diophantine equation $A{a}^p+B{b}^p=C{c}^3$ over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate $S$-unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation $a^p+d{b}^p=c^3$ over $K=\mathbb{Q}(\sqrt{-d})$ with $d\in \{7,19,43,67\}$, we determine an explicit bound (depending on $d$) such that no solutions of a certain type exist whenever $p$ exceeds this bound.