A priori estimates for semistable solutions of p -Laplace equations with general nonlinearity

Abstract

In this paper we consider the p-Laplace equation $-{\mathrm{\Delta }}_{\mathit{p}}\mathit{u}=\mathit{\lambda }\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}$ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{N}}$ with zero Dirichlet boundary condition, where $\mathit{p}\mathrm{>}1$ , $\mathit{\lambda }\mathrm{>}0$ and $\mathit{f}:[0\mathrm{,}\infty \mathrm{)}\to \mathbb{R}$ is a ${\mathit{C}}^1$ function with $\mathit{f}\mathrm{(}0\mathrm{)}\mathrm{>}0$ , ${\mathit{f}}^{\prime }⩾0$ and ${lim}_{\mathit{t}\to \infty }\frac{\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}}{{\mathit{t}}^{\mathit{p}-1}}=\infty$ . For the sequence ${\mathrm{(}{\mathit{u}}_{\mathit{\lambda }}\mathrm{)}}_{0\mathrm{<}\mathit{\lambda }\mathrm{<}{\mathit{\lambda }}^{\ast }}$ of minimal semi-stable solutions, by applying the semi-stability inequality we find a class of functions E that asymptotically behave like a power of f at infinity and show that $\Vert \mathit{E}\mathrm{(}{\mathit{u}}_{\mathit{\lambda }}\mathrm{)}{\Vert }_{{\mathit{L}}^1\mathrm{(}\mathrm{\Omega }\mathrm{)}}$ is uniformly bounded for $\mathit{\lambda }\mathrm{<}{\mathit{\lambda }}^{\ast }$ . Then using elliptic regularity theory we provide some new ${\mathit{L}}^{\infty }$ estimates for the extremal solution ${\mathit{u}}^{\ast }$ , under some suitable conditions on the nonlinearity f, where the obtained results require neither the convexity of f nor the strictly convexity of the domain. In particular, under some mild assumptions on f we show that ${\mathit{u}}^{\ast }\in {\mathit{L}}^{\infty }\mathrm{(}\mathrm{\Omega }\mathrm{)}$ for $\mathit{N}\mathrm{<}\mathit{p}+4\mathit{p}\mathrm{/}\mathrm{(}\mathit{p}-1\mathrm{)}$ , which is conjectured to be the optimal regularity dimension for ${\mathit{u}}^{\ast }$ .