On the solutions of $x^2=B{y}^p+C{z}^p$ and $2x^2=B{y}^p+C{z}^p$ over totally real fields

In this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2=B{y}^p+C{z}^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r\in \mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2=B{y}^p+2^r{z}^p$ ($r\in 1,2,4,5$) (resp., $2x^2=B{y}^p+2^r{z}^p$ with $r\in \mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$.