Non-trivial Solutions of $Aa^p+Bb^p=Cc^3$ 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.