Sharp estimates of radial minimizers of p-Laplace equations

TL;DR

This paper derives sharp pointwise estimates for semi-stable, radially symmetric solutions of p-Laplace equations, including extremal solutions, and constructs a wide class of unbounded solutions in higher dimensions.

Contribution

It provides new sharp estimates for radial solutions of p-Laplace equations and constructs numerous unbounded semi-stable solutions in certain dimensions.

Findings

Sharp pointwise estimates for solutions and derivatives

Optimal estimates for extremal solutions of nonlinear p-Laplace equations

Existence of many unbounded semi-stable solutions in high dimensions

Abstract

In this paper 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. As an application of these results, we obtain optimal pointwise estimates 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 $\lim_{t\rightarrow+\infty}{\frac{f(t)}{t^{p-1}}}=+\infty.$ In addition, we provide, for $N\geq p+4p/(p-1)$, a large family of semi-stable radially symmetric and decreasing unbounded $W^{1,p}(B_1)$ solutions.