Regularity of the extremal solution for singular p-Laplace equations

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 $n > p p'$

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

“…For a more general non-linearity, Cabré and Sanchón [5] proved that every semi-stable solution is bounded for a explicit exponent which is optimal for the boundedness of semi-stable solutions and, in particular, it is bigger than the critical Sobolev exponent p * − 1. For general h(s) and p > 1 the interested reader can see [4,10,41,43] for more regularity results about the extremal solution. In [4], Cabré, Capella and Sanchón treated the delicate issue about regularity of extremal solutions u * of (1.7) at λ = λ * when Ω is the unit ball of R N .…”