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

Abstract

In this paper, we study global positive $C^{2N}$-solutions of the geometrically interesting equation $(-\Delta)^N u + u^{-(4N-1)}= 0$ in $\mathbf R^{2N-1}$. We prove that any $C^{2N}$-solution $u$ of the equation having linear growth at infinity must satisfy the integral equation \[ u(x) = c_0 \int_{\mathbf R^{2N-1}} {|x - y|{u^{-(4N-1)}}(y)dy} \] for some positive constant $c_0$ and hence takes the following form \[ u(x) = (1+|x|^2)^{1/2} \] 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 $\mathbf R^{2N-1}$.