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

TL;DR

This paper classifies entire solutions with linear growth at infinity for a high-order nonlinear PDE involving the fractional Laplacian and inverse power nonlinearity, showing solutions are essentially quadratic in form.

Contribution

It proves that solutions with linear growth are characterized by a specific integral representation and are essentially quadratic functions, providing a classification and non-existence results.

Findings

Solutions with linear growth are of the form (1+|x|^2)^{1/2}

Such solutions satisfy a specific integral equation

Non-existence results for certain related equations

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 $\mathbf R^{2N-1}$.