Behavior near the origin of $f'(u^\ast)$ in radial singular extremal solutions

TL;DR

This paper investigates the behavior of the derivative of the extremal solution near zero in a semilinear elliptic problem, addressing an open question about singular solutions in the context of nonlinear PDEs.

Contribution

It provides new insights into the behavior of $f'(u^*)$ near zero for singular extremal solutions, solving an open problem posed by Brezis and Vázquez.

Findings

Characterization of $f'(u^*)$ near zero for singular solutions

Resolution of an open problem in elliptic PDE theory

Enhanced understanding of extremal solutions in nonlinear elliptic equations

Abstract

Consider the semilinear elliptic equation $-\Delta u=\lambda f(u)$ in the unit ball $B_1\subset \mathbb{R}^N$, with Dirichlet data $u|_{\partial B_1}=0$, where $\lambda\geq 0$ is a real parameter and $f$ is a $C^1$ positive, nondecreasing and convex function in $[0,\infty)$ such that $f(s)/s\rightarrow\infty$ as $s\rightarrow\infty$. In this paper we study the behavior of $f'(u^\ast)$ near the origin when $u^\ast$, the extremal solution of the previous problem associated to $\lambda=\lambda^\ast$, is singular. This answers to an open problems posed by Brezis and V\'azquez.