Asymptotic solutions of generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields K, i.e., Axᵖ+Byᵖ=Cz³ with prime exponent p and A,B,C ∈ OK \0\. For certain class of fields K, we prove that Axᵖ+Byᵖ=Cz³ has no asymptotic solutions over K (resp., solutions of certain type over K) with restrictions on $A,B,C$ (resp., for all A,B,C ∈ OK \0\). Finally, we present several local criteria over K.