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