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 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 >  pp′.