Regularity of the extremal solution for singular p-Laplace equations

Abstract

We study the regularity of the extremal solution $u^{\ast }$ to the singular reaction-diffusion problem $-{\Delta }_{p}u=\lambda f(u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $1<p<2$, $0<\lambda <{\lambda }^{\ast }$, $\Omega \subset {\mathbb{R}}^n$ 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 $u^{\ast }$, i.e. $u^{\ast }\in L^{\infty }(\Omega )$ if $n\le p+2$, $u^{\ast }\in L^{\frac{2n}{n-p-2}}(\Omega )$ if $n>p+2$ and $|\nabla u^{\ast }{|}^{p-1}\in L^{\frac{n}{n-(p^{\prime }+1)}}(\Omega )$ if $n>p{p}^{\prime }$.

References in corpus (1)

• Regularity of minimizers of semilinear elliptic problems up to dimension four