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 $\hbar=0$.
Categories: math.CV math.AP