Sharp estimates for semi-stable radial solutions of semilinear elliptic equations

TL;DR

This paper derives sharp pointwise estimates for semi-stable radial solutions of semilinear elliptic equations, providing optimal bounds for extremal solutions and their derivatives in the unit ball.

Contribution

It establishes the first sharp pointwise estimates for semi-stable radial solutions and applies these to obtain optimal bounds for extremal solutions of semilinear elliptic equations.

Findings

Sharp pointwise estimates for semi-stable radial solutions.

Optimal bounds for extremal solutions and derivatives.

Results applicable to a broad class of nonlinearities.

Abstract

This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-\Delta u=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. 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 semilinear elliptic equation $-\Delta u=\lambda f(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such that $f(s)/s\to\infty$ as $s\to\infty$.