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In this article, we study the solutions of certain type over $K$ of theDiophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is anodd 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 Diophantineequation $x^2= By^p+2^rz^p$ ($r\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with$r \in \mathbb{N}$) with prime exponent $p$, over $K$. We also present severalpurely local criteria of $K$.

On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields