*Published Paper*

**Inserted:** 22 apr 2016

**Last Updated:** 31 oct 2018

**Journal:** Manuscripta Mathematica

**Year:** 2015

**Abstract:**

We study the regularity of the extremal solution $u^*$ to the singular
reaction-diffusion problem $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u =0$ on
$\partial \Omega$, where $1<p<2$, $0 < \lambda < \lambda^*$, $\Omega \subset
\mathbb{R}^n$ is a smooth bounded domain and $f$ is any positive, superlinear,
increasing and (asymptotically) convex $C^1$ nonlinearity. We provide a simple
proof of known $L^r$ and $W^{1,r}$ \textit{a priori} estimates for $u^*$, i.e.
$u^* \in L^\infty(\Omega)$ if $n \leq p+2$, $u^* \in
L^{\frac{2n}{n-p-2}}(\Omega)$ if $n > p+2$ and $

\nabla u^*

^{p-1} \in
L^{\frac{n}{n-(p'+1)}} (\Omega)$ if $n > p p'$.