On the regularity and partial regularity of extremal solutions of a Lane-Emden system

Abstract

In this paper, we consider the system $-\Delta u =\lambda (v+1)^p,\;\;-\Delta v = \gamma (u+1)^\theta$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^N$ with the Dirichlet boundary condition $u=v=0$ on $\partial \Omega.$ Here $ \lambda,\gamma$ are positive parameters. Let $x_0$ be the largest root of the polynomial \begin{equation*} H(x) = x^4 - \frac{16p\theta(p+1)(\theta+1)}{(p\theta-1)^2}x^2 + \frac{16p\theta(p+1)(\theta+1)(p+\theta+2)}{(p\theta-1)^3}x -\frac{16p\theta(p+1)^2(\theta+1)^2}{(p\theta-1)^4}. \end{equation*} We show that the extremal solutions associated to the above system are bounded provided $N<2+2x_0.$ This improves the previous work in \cite{co1}. We also prove that, if $N\geq 2+2x_0,$ then the singular set of any extremal solution has Hausdorff dimension less or equal to $N-(2+2x_0).$