Classification of entire solutions of $(-\mathrm{\Delta }{)}^{N}u+u^{-(4N-1)}=0$ with exact linear growth at infinity in ${\mathbf{R}}^{2N-1}$

In this paper, we study global positive $C^{2N}$-solutions of the geometrically interesting equation $(-\mathrm{\Delta }{)}^{N}u+u^{-(4N-1)}=0$ in $\mathbf R^{2N-1}$\mathrm{.}Weprovethatany$C^{2N}$-solution$u$ of the equation having linear growth at infinity must satisfy the integral equation $$u(x)=c_0{\int }_{{\mathbf{R}}^{2N-1}}\mathrm{\mid }x-y\mathrm{\mid }{u}^{-(4N-1)}(y)dy$$ for some positive constant $c_0$ and hence takes the following form $$u(x)=(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$$ in ${\mathbf{R}}^{2N-1}$ up to dilations and translations. We also provide several non-existence results for positive $C^{2N}$-solutions of $(-\Delta)^N u = u^{-(4N-1)}$in$\mathbf R^{2N-1}$.