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

Hajlaoui, Hatem

doi:10.1090/proc/13789

On the regularity and partial regularity of extremal solutions of a Lane–Emden system
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Proc. Amer. Math. Soc. 147 (2019), 1987-1998 Request permission

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 Dirichlet boundary condition $u=v=0$ on $\partial \Omega .$ Here $\lambda ,\gamma$ are positive parameters and $1 < p \le \theta$. Let $x_0$ be the largest root of the polynomial \begin{align*} 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{align*} We show that the extremal solutions associated to the above system are bounded provided $N<2+2x_0.$ This improves the previous work by Craig Cowan (2015). We also prove that if $N\geq 2+2x_0,$ then the singular set of any extremal solution has Hausdorff dimension less than or equal to $N-(2+2x_0).$

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