Asymptotic Fermat equation of signature $(r,r,p)$ over totally real fields

Let $K$ be a totally real number field and ${\mathcal{O}}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r,r,p)$, specifically $x^r+y^r=d{z}^p$ over $K$, where $r,p\ge 5$ are rational primes and odd $d\in {\mathcal{O}}_K\setminus \{0\}$. For a certain class of fields $K$, we first prove that the equation $x^r+y^r=d{z}^p$ has no asymptotic solution $(a,b,c)\in {\mathcal{O}}_K^3$ with $2\mathrm{\mid }c$. Then, we study the asymptotic solutions $(a,b,c)\in {\mathcal{O}}_K^3$ to the equation $x^5+y^5=d{z}^p$ with $2\nmid c$. We use the modular method to prove these results.