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

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.