Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $A{x}^p+B{y}^p=C{z}^3$ with prime exponent $p$ and $A,B,C\in {\mathcal{O}}_K\setminus \{0\}$. For certain class of fields $K$, we prove that $A{x}^p+B{y}^p=C{z}^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C\in {\mathcal{O}}_K\setminus \{0\}$). Finally, we present several local criteria over $K$.