Regularity of the extremal solution for singular p-Laplace equations

TL;DR

This paper investigates the regularity of the extremal solution to a singular p-Laplace reaction-diffusion problem, providing simplified proofs of known a priori estimates and conditions for boundedness based on the domain's dimension.

Contribution

It offers a straightforward proof of existing $L^r$ and $W^{1,r}$ estimates for the extremal solution in singular p-Laplace equations with superlinear nonlinearities.

Findings

$u^* otin L^ fty( Omega)$ if $n > p+2$

$u^* otin L^{rac{2n}{n-p-2}}( Omega)$ if $n > p+2$

$| abla u^*|^{p-1} otin L^{rac{n}{n-(p'+1)}} ( Omega)$ if $n > p p'$

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 $n > p p'$.