Positive supersolutions for the Lane-Emden system with inverse-square potentials

TL;DR

This paper investigates the nonexistence of positive supersolutions for a Lane-Emden system with inverse-square potentials, identifying sharp supercritical regions and constructing solutions in subcritical cases.

Contribution

It provides sharp criteria for nonexistence in supercritical regimes and constructs explicit supersolutions in subcritical regimes for the system.

Findings

Sharp nonexistence regions for supercritical parameters.

Existence of positive supersolutions in subcritical cases.

Analysis of blow-up behavior near the origin.

Abstract

In this paper, we study the nonexistence of positive supersolutions for the following Lane-Emden system with inverse-square potentials \begin{equation}\label{0} \left\{ \begin{array}{lll} -\Delta u+\frac{\mu_1}{|x|^2} u= v^p \quad {\rm in}\ \, \Omega\setminus\{0\},\\[2mm] -\Delta v+\frac{\mu_2}{|x|^2} v= u^q \quad {\rm in}\ \, \Omega\setminus\{0\} \end{array} \right. \end{equation} for suitable $p,q>0$, $\mu_1,\mu_2\geq -(N-2)^2/4$, where $\Omega$ is a smooth bounded domain containing the origin in $\mathbb{R}^N$ with $N\geq 3$. Precisely, we provide sharp supercritical regions of $(p,q)$ for the nonexistence of positive supersolutions to system (\ref{0}) in the cases $-(N-2)^2/4\leq \mu_1,\mu_2<0$ and $-(N-2)^2/4\leq \mu_1<0\leq \mu_2$. Due to the negative coefficients $\mu_1,\mu_2$ of the inverse-square potentials, an initial blowing-up at the origin could be derived and an iteration procedure could be applied in the supercritical case to improve the blowing-up rate until the nonlinearities are not admissible in some weighted $L^1$ spaces. In the subcritical case, we prove the existence of positive supersolutions for system (\ref{s 1.1}) by specific radially symmetric functions.