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 R2N-1. Using the sub poly-harmonic property for positive C-2N-solutions of the differential inequality (-Delta)(N)u < 0 in R2N-1, we prove that any C-2N-solution u of the equation having linear growth at infinity must satisfy the integral equation u(x) = integral(R2N-1) vertical bar x -y vertical bar(-(4N-1))(u)(y) dy up to a multiple constant and hence take the following form: u(x) = (1+vertical bar x vertical bar(2))(1/2) in R2N-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 R2N-1.