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{16pθ(p+1)(θ+1)}{(pθ-1)^2}x^2 + \frac{16pθ(p+1)(θ+1)(p+θ+2)}{(pθ-1)^3}x -\frac{16pθ(p+1)^2(θ+1)^2}{(pθ-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 $N\ge 2+2x_0,$ then the singular set of any extremal solution has Hausdorff dimension less or equal to $N-(2+2x_0).$