On the solutions of the generalized Fermat equation over totally real number fields

Let $K$ be a totally real number field and ${\mathcal{O}}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $A{x}^p+B{y}^p+C{z}^p=0$ over $K$ with prime exponent $p$, where $A,B,C\in {\mathcal{O}}_K\setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $A{x}^p+B{y}^p+C{z}^p=0$ has no asymptotic solution $(a,b,c)\in {\mathcal{O}}_K^3$ with $2\mathrm{\mid }abc$. Then, under some assumptions on $A,B,C$, we also prove that $A{x}^p+B{y}^p+C{z}^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $A{x}^p+B{y}^p+C{z}^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field.