On partial differential equations of Waring's-problem form in several complex variables

In this paper, we first consider the pseudoprimeness of meromorphic solutions $u$ to a family of partial differential equations (PDEs) $H(u_{z_1},u_{z_2},\ldots,u_{z_n})=P(u)$ of Waring's-problem form, where $H(z_1,z_2,\ldots,z_n)$ is a nontrivial homogenous polynomial of degree $\ell$ in $\mathbf{C}^n$ and $P(w)$ is a polynomial of degree $\hbar$ in $\mathbf{C}$ with all zeros distinct. Then, we study when these PDEs can admit entire solutions in $\mathbf{C}^n$ and further find these solutions for important cases including particularly $u^\ell_{z_1}+u^\ell_{z_2}+\cdots+u^\ell_{z_n}=u^\hbar$, which are (often said to be) PDEs of super-Fermat form if $\hbar=0,\ell$ and an eikonal equation if $\ell=2$ and $\hbar=0$.