Non-trivial Integer Solutions of $x^r+y^r=D{z}^p$

In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey–Mazur Conjecture and the Eichler–Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=D{z}^p$ do not have any non-trivial primitive integer solutions, where $r\ge 5$ is a fixed prime, whenever $p$ is large enough. For $r\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, we get the same result with only assuming the Weak Frey–Mazur Conjecture.