On the solutions of $x^p+y^p=2^rz^p$, $x^p+y^p=z^2$ over totally real fields

We study the non-trivial primitive solutions of a specific type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ with prime exponent $p$ and $r \in \mathbb N$, over a certain class of totally real fields $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal O_K$ for the equation $x^p+y^p=2^rz^p$ with $p$ prime. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal O_K$.