Sharp estimates of radial minimizers of $p$–Laplace equations

Abstract:

We study semi-stable, radially symmetric and decreasing solutions $u\in W^{1,p}(B_1)$ of $-\Delta _p u=g(u)$ in $B_1\setminus \{ 0\}$, where $B_1$ is the unit ball of $\mathbb {R}^N$, $p>1$, $\Delta _p$ is the $p-$Laplace operator and $g$ is a general locally Lipschitz function. We establish sharp pointwise estimates for such solutions, which do not depend on the nonlinearity $g$. By applying these results, sharp pointwise estimates are obtained for the extremal solution and its derivatives (up to order three) of the equation $-\Delta _p u=\lambda f(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, where the nonlinearity $f$ is an increasing $C^1$ function with $f(0)>0$ and $\lim _{t\rightarrow +\infty }{\frac {f(t)}{t^{p-1}}}=+\infty .$